Index stringlengths 1 5 | Challenge stringlengths 41 1.59k | Answer in Latex stringclasses 198
values | Answer in Sympy stringlengths 1 783 | Variation stringclasses 33
values | Source stringclasses 100
values | Category stringclasses 5
values |
|---|---|---|---|---|---|---|
101 | Compute up to degree 5 ($x^5$) the terms of the Maclaurin series of $ f(x) = A \cdot e^{B \cdot \sin(x)} $.
Assume A and B are real and positive. | \frac{1}{6} A \left(B^3-B\right) x^3+\frac{1}{2} A B^2 x^2+\frac{1}{120} A \left(B^5-10 B^3+B\right) x^5+\frac{1}{24} A
\left(B^4-4 B^2\right) x^4+A B x+A | A*B**2*x**2/2 + A*B*x + A + x**5*A*(B**5 - 10*B**3 + B)/120 + x**4*A*(B**4 - 4*B**2)/24 + x**3*A*(B**3 - B)/6 | Symbolic-2 | U-Math
sequences_series
1ccc052c-9604-4459-a752-98ebdf3e0764 | Series |
102 | Find the radius of convergence of the following series. Assume A,B,G,H are real and positive.
$ \sum_{n=1}^\infty \frac{B \cdot (2 \cdot n)! \cdot (A \cdot x)^n}{G \cdot (H \cdot n)^{2 \cdot n}} $ | \frac{e^2 \cdot H^2}{4 \cdot A} | (E**2 * H**2) / (4 * A) | Symbolic-4 | U-Math
sequences_series
ca5ffe7c-f495-43dc-a653-de477cabc185 | Series |
103 | Compute the first 6 nonzero terms of the Maclaurin series of $f(x) = G \cdot \left(\sin(A \cdot x) \cdot \cos\left(\frac{ B \cdot \pi }{ 4 }\right) + \cos(A \cdot x) \cdot \sin\left(\frac{ B \cdot \pi }{ 4 }\right)\right)$
Assume A,B,G are real and positive. | \frac{1}{120} A^5 G x^5 \cos \left(\frac{\pi B}{4}\right)+\frac{1}{24} A^4 G x^4 \sin \left(\frac{\pi
B}{4}\right)-\frac{1}{6} A^3 G x^3 \cos \left(\frac{\pi B}{4}\right)-\frac{1}{2} A^2 G x^2 \sin \left(\frac{\pi
B}{4}\right)+A G x \cos \left(\frac{\pi B}{4}\right)+ G \sin \left(\frac{\pi B}{4}\right) | G*(A**5*x**5*cos(B*pi/4) + 5*A**4*x**4*sin(B*pi/4) - 20*A**3*x**3*cos(B*pi/4) - 60*A**2*x**2*sin(B*pi/4) + 120*A*x*cos(B*pi/4) + 120*sin(B*pi/4))/120 | Symbolic-3 | U-Math
sequences_series
f89bd354-18c9-4f31-b91f-cf6421e24921 | Series |
104 | Compute the terms up to order 4 ($x^4$) of the Maclaurin series of $f(x) = G /cdot e^{A cdot x} \cdot \cos(B cdot x)$.
Assume A,B,G are real and positive. | G x^3 \left(\frac{A^3}{6}-\frac{A B^2}{2}\right)+\frac{1}{2} G x^2 \left(A^2-B^2\right)+G x^4 \left(\frac{A^4}{24}-\frac{A^2
B^2}{4}+\frac{B^4}{24}\right)+A G x+G | G*(4*A*x**3*(A**2 - 3*B**2) + 24*A*x + x**4*(A**4 - 6*A**2*B**2 + B**4) + 12*x**2*(A**2 - B**2) + 24)/24 | Symbolic-3 | U-Math
sequences_series
d1fe21df-ee7f-40c2-9655-6bd6a7a23ff1 | Series |
105 | Compute $\lim_{x \to 0}\frac{(2 \cos (J \cdot x)+4) \csc (J \cdot x)}{5 \cdot A \cdot (J \cdot x)^3}-\frac{6}{5 \cdot A \cdot (J \cdot x)^4}$
Assume A,J are real and positive. | \frac{1}{A 150} | 1/(150*A) | Symbolic-2 | U-Math
sequences_series
068e40ce-9108-4ef8-8ee5-0d1471ebbe43 | Limits |
106 | Evaluate
$ \lim_{x \to 0^+} A \cdot \left( \frac{ \tan\left( \frac{B \cdot x}{2} \right) }{ \frac{B \cdot x}{2} } \right)^{ \frac{3 \cdot J}{(B \cdot x)^2} } $
Assume A,B,J are real and positive. | $A /cdot e^{\frac{J}{4}}$ | A*e**(J/4) | Symbolic-3 | U-Math
differential_calc
363dd580-f1fc-4867-a6ef-db2a03139745 | Limits |
107 | Evaluate
$ \lim_{x \to {5 \cdot A} } \left( \frac{ 3 \cdot B \cdot x }{ x - 5 \cdot A }-\frac{ 3 \cdot B }{ \ln\left(\frac{ x }{ 5 \cdot A }\right) } \right)^{J} $
Assume A,B,J are real and positive. | \left( \frac{3B}{2} \right)^J | (3*B/2)**J | Symbolic-3 | U-Math
differential_calc
2d799998-115a-489b-a48b-57090954303e | Limits |
108 | Evaluate
$ \lim_{x \to \infty} \left(A \cdot x - A \cdot J \cdot x^2 \cdot \ln\left(1 + \frac{ 1 }{ J \cdot x }\right)\right)^{B} $
Assume A,B,J are real and positive. | \left(\frac{A}{2 J}\right)^{B} | (A/(2*J))**B | Symbolic-3 | U-Math
differential_calc
efdc4110-cf56-4f37-bf54-40fdd5d58145 | Limits |
109 | Evaluate
$ \lim_{x \to 0^+} J \cdot \left( \frac{\tan(A \cdot x)}{A \cdot x} \right)^{\frac{ H }{3 \cdot B \cdot x^2}} $
Assume A,B,J,H are real and positive. | J e^{\frac{A^{2} H}{9 B}} | J*e**((A**2*H)/((9*B))) | Symbolic-4 | U-Math
differential_calc
99a2304d-5d8e-4245-90da-a80651ca15d8 | Limits |
110 | Evaluate
$ \lim_{x \to 0} \left| J \cdot \left( \frac{-\sin(A \cdot x)}{A \cdot x} \right)^{\frac{1}{4 \cdot B \cdot x^2}} \right| $
Assume A,B,J are real and positive. | \left( e^{-\frac{A^2}{6B}} \right)^{\frac{1}{4}} \left| J \right| | (exp(-A**2 / (6 * B)))**Rational(1, 4) * Abs(J) | Symbolic-3 | U-Math
differential_calc
84c6a419-c103-41d5-aad5-dd8e690c6e88 | Limits |
111 | Integrate
$ \int B \cdot \sin(J \cdot x)^4 \cdot \cos(J \cdot x)^6 dx $
Assume B,J are real and positive. | \frac{\frac{B \sin{\left(2 J x \right)}}{512} - \frac{B \sin{\left(4 J x \right)}}{256} - \frac{B \sin{\left(6 J x \right)}}{1024} + \frac{B \sin{\left(8 J x \right)}}{2048} + \frac{B \sin{\left(10 J x \right)}}{5120} + \frac{J x \left(256 A + 3 B\right)}{256}}{J} | (3*B*x)/256 + (B*sin(2*J*x))/(512*J) - (B*sin(4*J*x))/(256*J) - (B*sin(6*J*x))/(1024*J) + (B*sin(8*J*x))/(2048*J) + (B*sin(10*J*x))/(5120*J) | Symbolic-2 | U-Math
integral_calc
0c0ba3db-1470-4c36-975c-91ff5f51986f | Integrals |
112 | Solve the following integral. Assume A,B,G,H are real and positive.
$ \int \frac{A \cdot \sqrt[5]{x} + B \cdot x^{4/5} + H \cdot x^{6/5}}{x \left(G+x^{2/5}\right)} dx $ | \frac{5}{4} \left(\frac{4 A \tan ^{-1}\left(\frac{\sqrt[5]{x}}{\sqrt{G}}\right)}{\sqrt{G}}+2 x^{2/5} (B-H G)+2 G (H G-B) \log
\left(G+x^{2/5}\right)+H x^{4/5}\right) | (5/4)*(H*x**(4/5) + ((2*(x**(2/5)*(B - H*G)) + (4*(A*atan(x**(1/5)/(sqrt(G)))))/(sqrt(G))) + 2*(G*(-B + H*G)*log(G + x**(2/5), E)))) | Symbolic-4 | U-Math
integral_calc
126c4165-b3d5-4470-8412-08e79d9821cf | Integrals |
113 | Solve the following integral. Assume A,B,J are real and positive.
$ \int \frac{A }{ B \cdot \sin ^7(J \cdot x) \cdot \cos (J \cdot x)} dx $ | -\frac{A \left(2 \csc ^6(J x)+3 \csc ^4(J x)+6 \csc ^2(J x)+12 (\log (\cos (J x))-\log (\sin (J x)))\right)}{12 B J} | -A*(-12*log(sin(J*x)) + 12*log(cos(J*x)) + 2*(1/(sin(J*x)))**6 + 3*(1/(sin(J*x)))**4 + 6*(1/(sin(J*x)))**2)/(12*B*J) | Symbolic-3 | U-Math
integral_calc
00f6affb-905a-4109-a78e-2dde7a0b83accf | Integrals |
114 | Solve the following integral. Assume A,B,J are real and positive.
$ \int -\frac{2 \cdot A \cdot \sqrt{4 \cdot B + (J \cdot x)^2}}{ (J \cdot x)^4} dx $ | \frac{A \left(4 B+J^2 x^2\right)^{3/2}}{6 B J^4 x^3} | A*(4*B + J**2*x**2)**(3/2)/(6*B*J**4*x**3) | Symbolic-3 | U-Math
integral_calc
05ea9929-8cbb-432b-bbbb-ec1e74c9f401 | Integrals |
115 | Solve the following integral. Assume A,B,J are real and positive.
$ \int \left(\frac{B \cdot (4 \cdot A + J \cdot x)}{J \cdot x - 4 \cdot A}\right)^{\frac{ 3 }{ 2 }} dx $ | \frac{B \sqrt{\frac{B (4 A+J x)}{J x-4 A}} \left(\sqrt{4 A+J x} (J x-20 A)+24 A \sqrt{J x-4 A} \tanh ^{-1}\left(\frac{\sqrt{4
A+J x}}{\sqrt{J x-4 A}}\right)\right)}{J \sqrt{4 A+J x}} | B*sqrt(-B*(4*A + J*x)/(4*A - J*x))*(24*A*sqrt(-4*A + J*x)*atanh(sqrt(4*A + J*x)/sqrt(-4*A + J*x)) + (-20*A + J*x)*sqrt(4*A + J*x))/(J*sqrt(4*A + J*x)) | Symbolic-3 | U-Math
integral_calc
08c72d46-1abd-49e1-9c9c-ce509902be6e | Integrals |
116 | Solve the following integral. Assume A,B,J,G are real and positive.
$ \int \frac{ -A }{B \cdot (J \cdot x)^2 \cdot \left(3 \cdot G + (J \cdot x)^3\right)^{\frac{ 5 }{ 3 }} } dx $ | \frac{A \left(J^3 x^3+2 G\right)}{6 B J^2 G^2 x \left(J^3 x^3+3 G\right)^{2/3}} | A*(J**3*x**3 + 2*G)/(6*B*J**2*G**2*x*(J**3*x**3 + 3*G)**(2/3)) | Symbolic-4 | U-Math
integral_calc
4c1292e1-d4b3-4acf-afaf-eaac62f2662d | Integrals |
117 | Solve the following integral. Assume A,B,J,G,H are real and positive.
$ \int \frac{\sqrt{4 \cdot A \cdot x - 5 \cdot B} + 4 \cdot J \cdot x}{5 \cdot G \cdot \sqrt[4]{4 \cdot A \cdot x - 5 \cdot B} + H \cdot (4 \cdot A \cdot x - 5 \cdot B)^{\frac{3}{4}}} dx $ | \frac{\frac{\sqrt{H} \left(20 A^2 H^2 x+375 J G^2 \sqrt{4 A x-5 B}+5 B H \left(12 J H \sqrt{4 A x-5 B}-5 A H+25 J G\right)+A H
\left(12 J H x \sqrt{4 A x-5 B}-75 G \sqrt{4 A x-5 B}-100 J G x\right)\right)}{\sqrt[4]{4 A x-5 B}}-75 \sqrt{5} \sqrt{G}
\left(-A G H+B J H^2+5 J G^2\right) \tan ^{-1}\left(\frac{\sqrt{H}... | (75*sqrt(5)*sqrt(G)*(4*A*x - 5*B)**(1/4)*(A*G*H - B*J*H**2 - 5*J*G**2)*atan(sqrt(5)*sqrt(H)*(4*A*x - 5*B)**(1/4)/(5*sqrt(G))) + sqrt(H)*(20*A**2*H**2*x + A*H*(-100*J*G*x + 12*J*H*x*sqrt(4*A*x - 5*B) - 75*G*sqrt(4*A*x - 5*B)) + 5*B*H*(-5*A*H + 25*J*G + 12*J*H*sqrt(4*A*x - 5*B)) + 375*J*G**2*sqrt(4*A*x - 5*B)))/(15*A**2*... | Symbolic-5 | U-Math
integral_calc
147944c5-b782-48c5-a664-d66deb92d9a7 | Integrals |
118 | Solve the following integral. Assume A,B,J are real and positive.
$ \int \frac{3 \cdot A \cdot \csc ^7(2 \cdot J \cdot x) \sec (2 \cdot J \cdot x)}{ B } dx $ | -\frac{A \left(2 \csc ^6(2 J x)+3 \csc ^4(2 J x)+6 \csc ^2(2 J x)+12 (\log (\cos (2 J x))-\log (\sin (2 J x)))\right)}{8 B J} | -A*(-12*log(sin(2*J*x)) + 12*log(cos(2*J*x)) + 2*csc(2*J*x)**6 + 3*csc(2*J*x)**4 + 6*csc(2*J*x)**2)/(8*B*J) | Symbolic-3 | U-Math
integral_calc
1db212f0-2fac-410d-969d-fe3b5b55d076 | Integrals |
119 | Solve the following integral. Assume A,J are real and positive.
$ \int \frac{ A }{ (\sin(8 \cdot J \cdot x))^5 } dx $ | -\frac{A \left(\csc ^4(4 J x)+6 \csc ^2(4 J x)-\sec ^4(4 J x)-6 \sec ^2(4 J x)+24 (\log (\cos (4 J x))-\log (\sin (4 J
x)))\right)}{512 J} | -A*(-24*log(sin(4*J*x)) + 24*log(cos(4*J*x)) + csc(4*J*x)**4 + 6*csc(4*J*x)**2 - sec(4*J*x)**4 - 6*sec(4*J*x)**2)/(512*J) | Symbolic-2 | U-Math
integral_calc
275f7ceb-f331-4a3f-96ec-346e6d81b32a | Integrals |
120 | Solve the following integral. Assume A,B,J are real and positive.
$ \int \cos (2 \cdot J \cdot x) \left(A \cdot (J \cdot x)^3+3 \cdot B \right) dx $ | \frac{2 \sin (2 J x) \left(A J x \left(2 J^2 x^2-3\right)+6 B\right)+3 A \left(2 J^2 x^2-1\right) \cos (2 J x)}{8 J} | ((2*A*J*x*(2*J**2*x**2 - 3) + 12*B)*sin(2*J*x) + 3*A*(2*J**2*x**2 - 1)*cos(2*J*x))/(8*J) | Symbolic-3 | U-Math
integral_calc
47a11349-0386-4969-9263-d3cdfcc98cb9 | Integrals |
121 | Use factoring to calculate the following limit.
Assume A,B,J are real and positive.
$ \lim_{x \rightarrow \frac{K}{J}} \frac{(J \cdot x)^{4 \cdot B} - K^{4 \cdot B}}{A \cdot \left((J \cdot x)^{5 \cdot B}- K^{5 \cdot B}\right)} $ | \frac{4 K^{-B}}{5 A} | 4/(5*A*K**B) | Symbolic-3 | UGMathBench
Calculus_-_single_variable_0016 | Limits |
122 | Calculate the following limit.
Assume A,B,J are real and positive.
$ \lim_{x \to 0} \frac{B - B \cdot \cos (10 \cdot J \cdot x)}{A \cdot \cos ^2(6 \cdot J \cdot x) - A} $ | -\frac{25 B}{18 A} | -25*B/(18*A) | Symbolic-3 | UGMathBench
Calculus_-_single_variable_0022 | Limits |
123 | Calculate the following limit.
Assume A,B,J are real and positive.
$ \lim_{x\to 1} \frac{A \cdot (J \cdot x)^2 + 11 \cdot A \cdot J \cdot x - 12 \cdot A}{B \cdot \log (J \cdot x)} $ | \frac{13 A}{B} | 13*A/B | Symbolic-3 | UGMathBench
Calculus_-_single_variable_0508 | Limits |
124 | Calculate the following limit.
Assume A,J are real and A>1.
$ \lim\limits_{x\to+\infty} 4^{-\frac{1}{J \cdot x}} \left(\frac{(4 \cdot A)^{J \cdot x} + (6 \cdot A)^{J \cdot x} }{ A }\right)^{\frac{1}{J \cdot x}} $ | 6 A | 6*A | Symbolic-2 | UGMathBench
Calculus_-_single_variable_0512 | Limits |
125 | Calculate the following integral.
Assume A,B, J are real and positive.
$\int_{2 \cdot B}^{\infty} 3 \cdot A \cdot (J \cdot x)^2 e^{- (J \cdot x)^3} dx$ | \frac{A e^{-8 B^3 J^3}}{J} | A/(J*e**(8*B**3*J**3)) | Symbolic-3 | UGMathBench
Calculus_-_single_variable_0592 | Integrals |
126 | Evaluate the indefinite integral.
Assume A, J are real and positive.
$\int A \cdot \tan ^3(J \cdot x) \sec ^9(J \cdot x) dx$ | \frac{A \sec ^9(J x) \left(9 \sec ^2(J x)-11\right)}{99 J} | A*(9*sec(J*x)**2 - 11)*sec(J*x)**9/(99*J) | Symbolic-2 | UGMathBench
Calculus_-_single_variable_0604 | Integrals |
127 | Evaluate the indefinite integral.
Assume A, J are real and positive.
$\int 208 \cdot A \cdot \cos ^4(16 \cdot J \cdot x) dx$ | \frac{13 A (192 J x+8 \sin (32 J x)+\sin (64 J x))}{32 J} | 13*A*(192*J*x + 8*sin(32*J*x) + sin(64*J*x))/(32*J) | Symbolic-2 | UGMathBench
Calculus_-_single_variable_0606 | Integrals |
128 | Evaluate the integral.
Assume A, B, J, G are real and positive.
$ \int \frac{-38 \cdot A+10 \cdot B \cdot (J \cdot x)^2 - 48 \cdot J \cdot G \cdot x}{(J \cdot x)^3 - 5 (J \cdot x)^2 - 8 \cdot J \cdot x+48} dx $ | \frac{2 \left(\frac{7 (19 A-80 B+96 G)}{J x-4}+(19 A+200 B-72 G) \log (4-J x)+(-19 A+45 B+72 G) \log (J x+3)\right)}{49 J} | 2*(133*A - 560*B + 672*G + (J*x - 4)*((-19*A + 45*B + 72*G)*log(J*x + 3) + (19*A + 200*B - 72*G)*log(-J*x + 4)))/(49*J*(J*x - 4)) | Symbolic-4 | UGMathBench
Calculus_-_single_variable_0612 | Integrals |
129 | Evaluate the integral.
Assume A,B,J are real and positive.
$ \int A \cdot e^{J \cdot x} \sqrt{64 \cdot B-e^{2 \cdot J \cdot x}} dx$ | \frac{A \left(e^{J x} \sqrt{64 B-e^{2 J x}}+64 B \tan ^{-1}\left(\frac{e^{J x}}{\sqrt{64 B-e^{2 J x}}}\right)\right)}{2 J} | A*(64*B*atan(e**(J*x)/sqrt(64*B - e**(2*J*x))) + e**(J*x)*sqrt(64*B - e**(2*J*x)))/(2*J) | Symbolic-3 | UGMathBench
Calculus_-_single_variable_0624 | Integrals |
130 | Evaluate the following limit.
Assume A,B,J are real and positive.
$\lim_{x \to 0} \frac{-\frac{9}{2} \cdot B^2 (J \cdot x)^6 + 3 \cdot B \cdot J^3 \cdot x^3 + e^{-3 \cdot B \cdot (J \cdot x)^3}-1}{12 \cdot A \cdot (J \cdot x)^9} $ | -\frac{3 B^3}{8 A} | -3*B**3/(8*A) | Symbolic-3 | UGMathBench
Calculus_-_single_variable_0939 | Limits |
131 | Solve the following first-order differential equation:
Assume A,B,J,G are real and positive.
$ A \cdot \frac{dy}{dx} + 2 \cdot B \cdot y = J \cdot e^{-x}, \quad y(0) = G .$ | \frac {e^{-\frac{2 B x}{A}} \left(J \left(-e^{x \left(\frac{2 B}{A}-1\right)}\right)+A G-2 B
G+J\right)}{A-2 B} | (A*G - 2*B*G + J + J*(-e**x*((-A + 2*B)/A)))/(e**(2*B*x/A)*(A - 2*B)) | Symbolic-4 | MathOdyssey
Problem 340 from Differential Equations - College Math | Differential Equations |
132 | Consider the differential equation
$A \cdot \frac{dy}{dx} = B \cdot x \cdot y$.
Find the value of $y(\sqrt{2})$ given that $y(0) = 2 \cdot J$.
Assume A,B,J are real and positive. | 2 J e^{\frac{B}{A}} | 2*J*e**(B/A) | Symbolic-3 | MathOdyssey
Problem 339 from Differential Equations - College Math | Differential Equations |
133 | Evaluate the following limit:
$ \lim_{x \to \infty} \sqrt{-B + H \cdot (J \cdot x)^2 + 2 \cdot J \cdot G \cdot x}-\sqrt{3 \cdot A + H \cdot (J \cdot x)^2} $
Assume A,B,J,G,H are real and positive. | \frac{G}{\sqrt{H}} | G/sqrt(H) | Symbolic-5 | MathOdyssey
Problem 315 from Calculus and Analysis - College Math | Limits |
134 | Evaluate $\lim\limits_{x\to \frac{4 \cdot B}{ J }} \frac{A \cdot (J \cdot x - 4 \cdot B)}{\sqrt{J \cdot x}-2 \sqrt{ B }} $.
Assume A,B,J are real and positive. | 4 A \sqrt{B} | 4*A*sqrt(B) | Symbolic-3 | MathOdyssey
Problem 317 from Calculus and Analysis - College Math | Limits |
135 | Evaluate $\int_0^{4 \cdot B} (2 \cdot A \cdot x - \sqrt{(4 \cdot B \cdot J)^2 - (J \cdot x)^2}) dx$.
Assume A,B,J are real and positive. | 4 B^2 (4 A-\pi J) | 4*B**2*(4*A - J*pi) | Symbolic-3 | MathOdyssey
Problem 325 from Calculus and Analysis - College Math | Integrals |
136 | Evaluate the series $\sum\limits_{x=1}^\infty \frac{ A }{B \cdot (J + x) (J +x+2)} $.
Assume A,B,J are real and positive. | \frac{A (2 J+3)}{2 B (J+1) (J+2)} | A*(2*J + 3)/(2*(J + 2)*B*(J + 1)) | Symbolic-3 | MathOdyssey
Problem 326 from Calculus and Analysis - College Math | Series |
137 | Evaluate the limit $\lim\limits_{x \to 0} \frac{(A \cdot x+1)^{\frac{1}{A \cdot x}}-e}{B \cdot x} $.
Assume A,B are real and positive. | -\frac{e A}{2 B} | -A*e/(2*B) | Symbolic-2 | MathOdyssey
Problem 327 from Calculus and Analysis - College Math | Limits |
138 | Evaluate the series $\sum\limits_{n=0}^\infty \frac{ \left(\frac{1}{2 \cdot B}\right)^{A \cdot (2 n+1)}}{J \cdot (2 n+1)} $.
Assume A,B,J are real and positive. | \frac{\tanh ^{-1}\left(2^{-A} \left(\frac{1}{B}\right)^A\right)}{J} | atanh((1/(2*B))**A)/J | Symbolic-3 | MathOdyssey
Problem 328 from Calculus and Analysis - College Math | Series |
139 | Evaluate the limit
$\lim\limits_{n\to\infty}\sum\limits_{k=0}^{n-1}\frac{ A }{B \cdot \sqrt{J \cdot n^2-k^2}}$
Assume A,B, J are real and positive - and $J \ge 1$. | \frac{A}{B} \arcsin\left(\frac{1}{\sqrt{J}}\right) | A*asin(1/sqrt(J))/B | Symbolic-3 | MathOdyssey
Problem 329 from Calculus and Analysis - College Math | Limits |
140 | Evaluate the iterated integral $\int_0^1dy\int_y^1 e^{-A \cdot (J \cdot x)^2} + B \cdot e^{J \cdot x} dx$.
Assume A,B,J are real and positive. | \frac{2 A B \left(e^J (J-1)+1\right)-e^{-A J^2}+1}{2 A J^2} | (e**(A*J**2)*(2*A*B*(e**J*(J - 1) + 1) + 1) - 1)/(2*A*J**2*e**(A*J**2)) | Symbolic-3 | MathOdyssey
Problem 336 from Calculus and Analysis - College Math | Integrals |
141 | What is the integral of $ 2 \cdot A \cdot x - B \cdot x^{7 \cdot J} \tan ^{-1}(3 \cdot G) $
Assume A,B,J,G are real and positive. | x \left(A x-\frac{B x^{7 J} \tan ^{-1}(3 G)}{7 J+1}\right) | x*((A*x*(7*J + 1) - B*x**(7*J)*atan(3*G))/(7*J + 1)) | Symbolic-4 | GHOSTS
Symbolic Integration
Q97 | Integrals |
142 | What is the integral of
$ A + B \cdot J \cdot x + \cosh (2 \cdot G) \cdot (J \cdot x)^{3 \cdot H} $
Assume A,B,J,G,H are real and positive. | A x+\frac{1}{2} B J x^2+\frac{x \cosh (2 G) (J x)^{3 H}}{3 H+1} | x*(2*(J*x)**(3*H)*cosh(2*G) + (2*A + B*J*x)*(3*H + 1))/(2*(3*H + 1)) | Symbolic-5 | GHOSTS
Symbolic Integration
Q98 | Integrals |
143 | What is the integral of $12 \cdot A + 6 \cdot B \cdot \cosh (J \cdot x)$
Assume A,B,J are real and positive. | 12 A x+\frac{6 B \sinh (J x)}{J} | 12*A*x + 6*B*sinh(J*x)/J | Symbolic-3 | GHOSTS
Symbolic Integration
Q90 | Integrals |
144 | What is the integral of
$ 4 \cdot (B \cdot x)^{7 \cdot J} + G \cdot \sin (H + A \cdot x) $
Assume A,B,J,G,H are real and positive. | \frac{4 A x (B x)^{7 J}-(7 J+1) G \cos (H+A x)}{7 J A+A} | (4*A*x*(B*x)**(7*J) - G*(7*J + 1)*cos(H + A*x))/(A*(7*J + 1)) | Symbolic-5 | GHOSTS
Symbolic Integration
Q14 | Integrals |
145 | What is the integral of
$ 2 x + 2 \cdot B \cdot x^{2 \cdot J}+\frac{x}{G \cdot x + H \cdot x \cdot e^{A \cdot x}} $.
Assume A,B,J,G,H are real and positive. | x \left(x+\frac{2 B x^{2 J}}{2 J+1}\right)-\frac{\log \left(G A \left(G+H e^{A x}\right)\right)}{G A}+\frac{\log \left(e^{A
x}\right)}{G A} | (G*A*x*((x*(2*J + 1) + 2*B*x**(2*J))/(2*J + 1)) + log(e**(A*x)) - log(G*A*(G + H*e**(A*x))))/(G*A) | Symbolic-5 | GHOSTS
Symbolic Integration
Q7 | Integrals |
146 | What is the integral of
$ B \cdot \log (3 \cdot H \cdot x) \cos (J \cdot \log (\sin (3))) - A \cdot x $
Assume A,B,J,H are real and positive. | B x (\log (3 H x)-1) \cos (J \log (\sin (3)))-\frac{A x^2}{2} | -A*x**2/2 + B*x*(log(3*H*x) - 1)*cos(J*log(sin(3))) | Symbolic-4 | GHOSTS
Symbolic Integration
Q15 | Integrals |
147 | What is the integral of
$ 3 \cdot A \cdot x - 4 \cdot B \cdot (H \cdot x)^2 \cdot \cos (J \cdot x + 3 \cdot G) $
Assume A,B,J,G,H are real and positive. | \frac{3 A x^2}{2}-\frac{8 B H^2 x \cos (J x+3 G)}{J^2}-\frac{4 B H^2 \left(J^2 x^2-2\right) \sin (J x+3 G)}{J^3} | (3*A*J**3*x**2 - 16*B*J*H**2*x*cos(J*x + 3*G) - 8*B*H**2*(J**2*x**2 - 2)*sin(J*x + 3*G))/(2*J**3) | Symbolic-5 | GHOSTS
Symbolic Integration
Q18 | Integrals |
148 | What is the integral of
$ A \cdot \tan ^{-1}(B \cdot x) + J \cdot \log (G \cdot \tanh (3 \cdot H))-3 $
Assume A,B,J,G,H are real and positive. | -\frac{A \log \left(B^2 x^2+1\right)}{2 B}+A x \tan ^{-1}(B x)+x (J \log (G \tanh (3 H))-3) | A*x*atan(B*x) - A*log(B**2*x**2 + 1)/(2*B) + x*(J*log(G*tanh(3*H)) - 3) | Symbolic-5 | GHOSTS
Symbolic Integration
Q20 | Integrals |
149 | What is the integral of
$ A \cdot (J \cdot x + 4 \cdot G) \cdot (3 \cdot J \cdot x + 4 \cdot G) e^{B \cdot x \cdot (J \cdot x + 4 \cdot G)^2} $
Assume A,B,J,G are real and positive. | \frac{A e^{B x (J x+4 G)^2}}{B} | A*e**(B*x*(J*x + 4*G)**2)/B | Symbolic-4 | GHOSTS
Symbolic Integration
Q22 | Integrals |
150 | What is the integral of
$ -A \cdot e^{3 \cdot B \cdot x} \cdot \sin \left(J \cdot e^{3 \cdot B \cdot x}\right) $
Assume A,B,J are real and positive. | \frac{A \cos \left(J e^{3 B x}\right)}{3 B J} | A*cos(J*e**(3*B*x))/(3*B*J) | Symbolic-3 | GHOSTS
Symbolic Integration
Q29 | Integrals |
151 | If $\log_{(2 \cdot A)} x - 2 \cdot J \cdot \log _{(2 \cdot A)} y = 2 \cdot B$, determine $y$, as a function of $x$
Assume A,B,J are real and positive. | e^{\frac{\log (x)-2 B \log (2 A)}{2 J}} | e**((-2*B*log(2*A) + log(x))/(2*J)) | Symbolic-3 | OlympiadBench
oe_to_maths_en_comp
2498 | Differential Equations |
152 | If $f(x)=2 \cdot A \cdot x+ B $ and $g(f(x)) = 4 \cdot J \cdot x^{2}+ G$, determine an expression for $g(x)$.
Assume A,B,J,G are real and positive. | \frac{J (x-B)^2}{A^2}+G | G + J*(-B + x)**2/A**2 | Symbolic-4 | OlympicArena
Math_1381 | Series |
153 | Solve the following integral. Assume A,B,J are real and B>0.
$\int_0^{\frac{\pi}{2 \cdot J}} \frac{A \cdot x \cdot \sin(2 \cdot J \cdot x)}{B + \cos^2(2 \cdot J \cdot x)} dx$ | \frac{A\pi}{4J^2\sqrt{B}} \arctan\!\frac{1}{\sqrt{B}} | A*pi*atan(1/sqrt(B))/(4*sqrt(B)*J**2) | Symbolic-3 | OBMU 2019 - Q21 | Integrals |
154 | Solve the following integral. Assume A,B,J,G are real and positive.
$\int_{1}^{2} \frac{A \cdot e^{J \cdot x} \cdot (J \cdot x - 1)}{J \cdot x \cdot \left(B \cdot e^{J \cdot x} + J \cdot G \cdot x\right)} dx$ | -\frac{A \log \left(\frac{2 (e B+G)}{e^2 B+2 G}\right)}{B J} | -A*log((2*(B*E + G))/(B*E**2 + 2*G), E)/(B*J) | Symbolic-4 | OBMU 2019 - Q18 | Integrals |
155 | Solve the following integral. Assume A,B,J are real and positive.
Solve the following integral:
$\int_{0}^{\pi} A \cdot \log(B \cdot (\sin(x))^{ J }) dx$ | A \pi \log\left(\frac{B}{2^J}\right) | A*(pi*log(B/(2**J), E)) | Symbolic-3 | OBMU 2019 - Q22 | Integrals |
156 | Evaluate the following hypergeometric function. Assume A,B are real numbers. Return a closed-form symbolic answer.
$ {}_2F_1\left( \begin{array}{c} 1 ,1 \\ 2 \end{array}; (-A)^{B} \right) $ | -(-A)^{-B} \log \left(1-(-A)^B\right) | log(1 - (-A)**B, E)/((-A)**B) | Symbolic-2 | ASyMOB
Hypergeometrics
Q1 | Hypergeometrics |
157 | Evaluate the following hypergeometric function. Assume the parameters: A,B are real numbers. Return a closed-form symbolic answer.
$ {}_2F_1\left( \begin{array}{c} 1 ,1 \\ 3 \end{array}; -2 \cdot (A^{B}) \right) $ | \frac{1}{2} A^{-2 B} \left(\left(2 A^B+1\right) \log \left(2 A^B+1\right)-2
A^B\right) | (-2*(A**B) + (2*(A**B) + 1)*log(2*(A**B) + 1, E))/(2*(A**B)**2) | Symbolic-2 | ASyMOB
Hypergeometrics
Q2 | Hypergeometrics |
158 | Evaluate the following hypergeometric function. Assume the parameters: A,B,G,H are real numbers. Return a closed-form symbolic answer.
$ {}_8F_7\left( \begin{array}{c} 1,1,1, A , B , 1, G, H \\ 2,2, H, G, B, 1, A \end{array}; -1 \right) $ | \frac{\pi ^2}{12} | pi**2/12 | Symbolic-4 | ASyMOB
Hypergeometrics
Q3 | Hypergeometrics |
159 | Evaluate the following hypergeometric function. Assume the parameters: x,A,B,G,H are real numbers. Return a closed-form symbolic answer.
$ {}_3F_2\left( \begin{array}{c} -1,-A, -B \\ -H, -G \end{array}; x \right) $ | 1-\frac{A B x}{H G} | 1 - A*(B*x)/(H*G) | Symbolic-4 | ASyMOB
Hypergeometrics
Q4 | Hypergeometrics |
160 | Solve the following integral. Assume the parameters: A,B,J are real numbers. Return a closed-form symbolic answer.
$ \int \frac{ A }{ B + (x \cdot J)^3 } dx $ | -\frac{A \left(\log \left(B^{2/3}-\sqrt[3]{B} J x+J^2 x^2\right)-2 \log \left(\sqrt[3]{B}+J x\right)+2 \sqrt{3} \tan
^{-1}\left(\frac{1-\frac{2 J x}{\sqrt[3]{B}}}{\sqrt{3}}\right)\right)}{6 B^{2/3} J} | -A*(-2*log(B**(1/3) + J*x) + log(B**(2/3) - B**(1/3)*J*x + J**2*x**2) + 2*sqrt(3)*atan(sqrt(3)*(B**(1/3) - 2*J*x)/(3*B**(1/3))))/(6*B**(2/3)*J) | Symbolic-3 | ASyMOB
Hypergeometrics
Q5 | Hypergeometrics |
161 | Solve the following integral. Assume A,B are positive integers.
$ \int \frac{(2 \cdot A + (2 \cdot A - B) \cdot x^{B}) \cdot x^{A - 1}}{2 \cdot (1 + x^{B} + x^{2 \cdot A}) \cdot \sqrt{1 + x^{B}}} dx $ | \tan ^{-1}\left(\frac{\left x^A}{\sqrt{x^B+1}}\right) | atan(x**A/(sqrt(x**B + 1))) | Symbolic-2 | ASyMOB
Hypergeometrics
Q6 | Hypergeometrics |
162 | Evaluate the following hypergeometric function. Assume the parameters: A,B,G are real numbers. Return a closed-form symbolic answer.
$ {}_2F_1\left( \begin{array}{c} A, G \\ A \end{array}; -B \right) $ | (B+1)^{-G} | (B+1)**(-G) | Symbolic-3 | ASyMOB
Hypergeometrics
Q7 | Hypergeometrics |
163 | Evaluate the following hypergeometric function. Assume the parameters: A,B are real numbers. Return a closed-form symbolic answer.
$ {}_1F_1\left( \begin{array}{c} A \\ A \end{array}; B \right) $ | e^B | E**B | Symbolic-2 | ASyMOB
Hypergeometrics
Q8 | Hypergeometrics |
164 | Evaluate the following hypergeometric function. Assume the parameters: A,B,G are real numbers. Return a closed-form symbolic answer.
$ {}_2F_1\left( \begin{array}{c} A, -2 \\ 2+G \end{array}; B \right) $ | -\frac{(-A-1) A B^2}{(G+2) (G+3)}-\frac{2 A B}{G+2}+1 | 1 - (2*A*B)/(2+G) + (B**2 * A * (A+1))/((2+G)*(3+G)) | Symbolic-3 | ASyMOB
Hypergeometrics
Q9 | Hypergeometrics |
165 | Evaluate the following hypergeometric function. Assume the parameters: A,B,G, H are real numbers. Return a closed-form symbolic answer.
$ {}_3F_1\left( \begin{array}{c} (A + 1), -2, (H + 3) \\ (2 + G) \end{array}; \frac{B}{2} \right) $ | \frac{(A+1) (A+2) B^2 (H+3) (H+4)}{4 (G+2) (G+3)}-\frac{(A+1) B (H+3)}{G+2}+1 | (-(A + 1)*B*(H + 3) / (G + 2) + ((( (B**2) * ((H + 3)*(H + 4)))*(A + 2))*(A + 1)) / ((4*((G + 2)*(G + 3)))) ) + 1 | Symbolic-4 | ASyMOB
Hypergeometrics
Q10 | Hypergeometrics |
166 | Solve the following differential equation:
$ B \cdot x \cdot y' + B \cdot y = A \cdot x \cdot \sin(x), y(\pi) = G $
Assume the parameters: A,B,G are real numbers. | \frac{A \sin (x)+A (-x) \cos (x)-\pi A+\pi B G}{B x} | (-A*pi + A*sin(x) + B*G*pi + A*(-x)*cos(x)) / (B*x) | Symbolic-3 | ASyMOB
Differential_Equations
Q1 | Differential Equations |
167 | Solve the following differential equation:
$ A \cdot y' = B \cdot e^{J \cdot x} \cdot y , y(H) = G $
Assume the parameters: A,B,G,H,J are real numbers. | G e^{\frac{B e^{J x}}{A J}-\frac{B e^{H J}}{A J}} | G*e**(B*(-e**(H*J) + e**(J*x))/(A*J)) | Symbolic-5 | ASyMOB
Differential_Equations
Q2 | Differential Equations |
168 | Solve the following differential equation:
$ A \cdot y' = 2 \cdot B \cdot x \cdot y^2 - G \cdot y , y(H) = J $
Assume the parameters: A,B,G,H,J are real numbers. | \frac{G^2 J e^{\frac{G H}{A}}}{-2 B G H J e^{\frac{G x}{A}}+2 B G J x e^{\frac{G
H}{A}}+2 A B J e^{\frac{G H}{A}}-2 A B J e^{\frac{G x}{A}}+G^2 e^{\frac{G
x}{A}}} | G**2*J*e**(G*H/A)/(2*A*B*J*e**(G*H/A) - 2*A*B*J*e**(G*x/A) - 2*B*G*H*J*e**(G*x/A) + 2*B*G*J*e**(G*H/A)*x + G**2*e**(G*x/A)) | Symbolic-5 | ASyMOB
Differential_Equations
Q3 | Differential Equations |
169 | Solve the following differential equation:
$ A \cdot y' = B \cdot x \cdot y^{G+1} - y , y(H) = J $
Assume the parameters: A,B,G,H,J are real numbers. | \left(\frac{J^{-G} e^{\frac{G (x-H)}{A}} \left(-A B J^G-B G H J^G+G\right)+A B+B G
x}{G}\right)^{-1/G} | ((B*J**G*(A + G*x) - e**(G*(-H + x) / A)*(A*B*J**G + B*G*H*J**G - G)) / (G*J**G))**(-1/G) | Symbolic-5 | ASyMOB
Differential_Equations
Q4 | Differential Equations |
170 | Solve the following differential equation:
$ A \cdot y' = \frac{(B+1) \cdot x \cdot y + G \cdot y^2}{x^2} , y(H) = J $
Assume the parameters: A,B,G,H,J are real numbers. | \frac{H J (A-B-1) x^{\frac{A+B+1}{A}}}{-G J x H^{\frac{B+1}{A}}+G H J
x^{\frac{B+1}{A}}+x (A-B-1) H^{\frac{A+B+1}{A}}} | H*x**((A + B + 1)/A)*J*(A - B - 1) / (G*H*J*x**((B + 1)/A) - G*H**((B + 1)/A)*J*x + H**((A + B + 1)/A)*x*(A - B - 1)) | Symbolic-5 | ASyMOB
Differential_Equations
Q5 | Differential Equations |
171 | Solve the following differential equation:
$ y' = A \cdot e^{-G \cdot x} - 2 \cdot B \cdot y , y(H) = J $
Assume the parameters: A,B,G,H,J are real and positive numbers. | \frac{e^{-2 B x} \left(A \left(-e^{H (2 B-G)}\right)+A e^{x (2 B-G)}-G J e^{2 B
H}+2 B J e^{2 B H}\right)}{2 B-G} | (A*e**(x*(2*B - G)) + 2*B*J*e**(2*B*H) - G*J*e**(2*B*H) + A*(-e**(H*(2*B - G)))) / (e**(2*B*x)*(2*B - G)) | Symbolic-5 | ASyMOB
Differential_Equations
Q6 | Differential Equations |
172 | Solve the following differential equation:
$ A \cdot y' = 3 \cdot x^2 \cdot \left( B + y^2 \right) , y(H) = J $
Assume the parameters: A,B,G,H,J are real and positive numbers. | \sqrt{B} \tan \left(\frac{A \tan ^{-1}\left(\frac{J}{\sqrt{B}}\right)-\sqrt{B}
H^3+\sqrt{B} x^3}{A}\right) | sqrt(B)*tan((A*atan(J/sqrt(B)) - sqrt(B)*H**3 + sqrt(B)*x**3)/A) | Symbolic-4 | ASyMOB
Differential_Equations
Q7 | Differential Equations |
173 | Solve the following differential equation:
$ A \cdot y' = \frac{2 \cdot G \cdot (A \cdot y + B \cdot x)}{x} , y(H) = J $
Assume the parameters: A,B,G,H,J are real and positive numbers. | -\frac{H^{-2 G} \left(A J x^{2 G}-2 A G J x^{2 G}+2 B G x H^{2 G}-2 B G H x^{2
G}\right)}{A (2 G-1)} | (2*A*G*J*x**(2*G) - A*J*x**(2*G) + 2*B*G*H*x**(2*G) - 2*B*G*H**(2*G)*x)/(H**(2*G)*A*(2*G - 1)) | Symbolic-5 | ASyMOB
Differential_Equations
Q8 | Differential Equations |
174 | Solve the following differential equation:
$ A \cdot y' = \frac{ (A \cdot y + B \cdot x)}{x} , y(H) = J $
Assume the parameters: A,B,H,J are real and positive numbers. | \frac{x (A J+B H \log (x)-B H \log (H))}{A H} | x*(A*J - B*H*log(H) + B*H*log(x))/(A*H) | Symbolic-4 | ASyMOB
Differential_Equations
Q9 | Differential Equations |
175 | Solve the following differential equation:
$ A \cdot y' = \left(\frac{x}{G}\right)^{B} , y(H) = J $
Assume the parameters: A,B,G,H,J are real and positive numbers. | \frac{G^{-B} \left(A J G^B+A B J G^B-H^{B+1}+x^{B+1}\right)}{A (B+1)} | (A*B*G**B*J + A*G**B*J - H**(B + 1) + x**(B + 1))/(G**B*A*(B + 1)) | Symbolic-5 | ASyMOB
Differential_Equations
Q10 | Differential Equations |
176 | Solve the following differential equation:
$ A \cdot y' = G \cdot x - 2 \cdot B \cdot x \cdot y , y(H) = J $
Assume the parameters: A,B,G,H,J are real and positive numbers. | \frac{e^{-\frac{B x^2}{A}} \left(G \left(-e^{\frac{B H^2}{A}}\right)+G e^{\frac{B
x^2}{A}}+2 B J e^{\frac{B H^2}{A}}\right)}{2 B} | (2*B*J*e**(B*H**2/A) + G*e**(B*x**2/A) + G*(-e**(B*H**2/A)))/(2*B*e**(B*x**2/A)) | Symbolic-5 | ASyMOB
Differential_Equations
Q11 | Differential Equations |
177 | Solve the following differential equation:
$ y' = 2 \cdot A \cdot \sin (G \cdot x) + 5 \cdot B \cdot \sin (2 x) - y , y(H - 1) = J - 1 $
Assume the parameters: A,B,G,H,J are real and positive numbers. | -\frac{e^{-x-1} \left(2 A e^H \sin (G (H-1))-2 A G e^H \cos (G (H-1))-2 A e^{x+1}
\sin (G x)+2 A G e^{x+1} \cos (G x)-2 B G^2 e^H \cos (2 (H-1))-2 B G^2 e^H \sin
(1-H) \cos (1-H)+2 B G^2 e^{x+1} \cos (2 x)-2 B G^2 e^{x+1} \sin (x) \cos (x)-2
B e^H \cos (2 (H-1))-2 B e^H \sin (1-H) \cos (1-H)+2 B e^{x+1} \cos (... | e**(-x - 1)*(2*A*G*e**H*cos(G*(H - 1)) - 2*A*G*e**(x + 1)*cos(G*x) - 2*A*e**H*sin(G*(H - 1)) + 2*A*e**(x + 1)*sin(G*x) - B*G**2*e**H*sin(2*H - 2) + 2*B*G**2*e**H*cos(2*H - 2) + B*G**2*e**(x + 1)*sin(2*x) - 2*B*G**2*e**(x + 1)*cos(2*x) - B*e**H*sin(2*H - 2) + 2*B*e**H*cos(2*H - 2) + B*e**(x + 1)*sin(2*x) - 2*B*e**(x + 1... | Symbolic-5 | ASyMOB
Differential_Equations
Q12 | Differential Equations |
178 | Solve the following differential equation:
$ y' = A \cdot \tan (B \cdot y) , y(H) = J $
Assume the parameters: A,B,H,J are real and positive numbers. | \frac{\sin ^{-1}\left(\sin (B J) e^{A B x-A B H}\right)}{B} | asin(e**(A*B*(-H + x))*sin(B*J))/B | Symbolic-4 | ASyMOB
Differential_Equations
Q13 | Differential Equations |
179 | Solve the following differential equation:
$ A \cdot y' = \sin^2(B \cdot y) + 2 \cdot \cos^2(B \cdot y) , y(H) = J $
Assume the parameters: A,B,H,J are real and positive numbers. | \frac{\tan ^{-1}\left(\sqrt{2} \tan \left(\frac{\frac{\sqrt{2} A \tan
^{-1}\left(\frac{\tan (B J)}{\sqrt{2}}\right)-2 B H}{\sqrt{2}}+\sqrt{2} B
x}{A}\right)\right)}{B} | atan(sqrt(2)*tan((A*atan(sqrt(2)*tan(B*J)/2) - sqrt(2)*B*H + sqrt(2)*B*x)/A))/B | Symbolic-4 | ASyMOB
Differential_Equations
Q14 | Differential Equations |
180 | Solve the following differential equation:
$ A \cdot y' = 2 \cdot G \cdot \cos ^2(B \cdot y) - \sin ^2(B \cdot y) , y(H - 1) = J - 1 $
Assume the parameters: A,B,G,H,J are real and positive numbers. | \frac{\tan ^{-1}\left(\sqrt{2} \sqrt{G} \tanh \left(\frac{\frac{\sqrt{2} A \tanh
^{-1}\left(\frac{\tan (B (J-1))}{\sqrt{2} \sqrt{G}}\right)-2 B \sqrt{G} H+2 B
\sqrt{G}}{\sqrt{2}}+\sqrt{2} B \sqrt{G} x}{A}\right)\right)}{B} | atan(sqrt(2)*sqrt(G)*tanh((A*atanh(sqrt(2)*tan(B*(J - 1))/(2*sqrt(G))) - sqrt(2)*B*sqrt(G)*H + sqrt(2)*B*sqrt(G)*x + sqrt(2)*B*sqrt(G))/A))/B | Symbolic-5 | ASyMOB
Differential_Equations
Q15 | Differential Equations |
181 | Solve the following differential equation:
$ G \cdot x + y'' + y' + y = 0 , y(0) = H, y'(0) = J $
Assume the parameters: G,H,J are real and positive numbers. | -\frac{1}{3} e^{-x/2} \left(-3 G e^{x/2}+3 G e^{x/2} x-\sqrt{3} G \sin
\left(\frac{\sqrt{3} x}{2}\right)+3 G \cos \left(\frac{\sqrt{3}
x}{2}\right)-\sqrt{3} H \sin \left(\frac{\sqrt{3} x}{2}\right)-3 H \cos
\left(\frac{\sqrt{3} x}{2}\right)-2 \sqrt{3} J \sin \left(\frac{\sqrt{3}
x}{2}\right)\right) | (-3*G*e**(x/2)*x + 3*G*e**(x/2) - 2*sqrt(3)*G*cos(sqrt(3)*x/2 + pi/6) + 2*sqrt(3)*H*sin(sqrt(3)*x/2 + pi/3) + 2*sqrt(3)*J*sin(sqrt(3)*x/2))/(3*e**(x/2)) | Symbolic-3 | ASyMOB
Differential_Equations
Q16 | Differential Equations |
182 | Solve the following differential equation:
$ A \cdot y'' + B \cdot y' - 6 \cdot G \cdot y = 0 , y(0) = H - 1, y'(0) = J $
Assume the parameters: A,B,G,H,J are real and positive numbers. | \frac{e^{-\frac{x \left(\sqrt{24 A G+B^2}+B\right)}{2 A}} \left(e^{\frac{x \sqrt{24
A G+B^2}}{A}} \left(H \sqrt{24 A G+B^2}-\sqrt{24 A G+B^2}+2 A J\right)+B (H-1)
\left(e^{\frac{x \sqrt{24 A G+B^2}}{A}}-1\right)+H \sqrt{24 A G+B^2}-\sqrt{24 A
G+B^2}-2 A J\right)}{2 \sqrt{24 A G+B^2}} | (-2*A*J + H*sqrt(24*A*G + B**2) + e**(x*sqrt(24*A*G + B**2)/A)*(2*A*J + H*sqrt(24*A*G + B**2) - sqrt(24*A*G + B**2)) + (e**(x*sqrt(24*A*G + B**2)/A) - 1)*B*(H - 1) - sqrt(24*A*G + B**2))/(2*e**(x*(B + sqrt(24*A*G + B**2))/(2*A))*sqrt(24*A*G + B**2)) | Symbolic-5 | ASyMOB
Differential_Equations
Q17 | Differential Equations |
183 | Solve the following differential equation:
$ G \cdot y + y'' = 0 , y(A) = H, y'(0) = J $
Assume the parameters: A,G,H,J are real and positive numbers. | \frac{\sec \left(A \sqrt{G}\right) \left(\sqrt{G} H \cos \left(\sqrt{G} x\right)-J
\sin \left(\sqrt{G} (A-x)\right)\right)}{\sqrt{G}} | (sqrt(G)*H*cos(sqrt(G)*x) - J*sin(sqrt(G)*(A - x)))*sec(A*sqrt(G))/sqrt(G) | Symbolic-4 | ASyMOB
Differential_Equations
Q18 | Differential Equations |
184 | Solve the following differential equation:
$ y'' - G \cdot y = 0 , y(A) = H, y'(B) = J $
Assume the parameters: A,B,G,H,J are real and positive numbers. | \frac{e^{-\sqrt{G} x} \left(\sqrt{G} H e^{\sqrt{G} (A+2 B)}-J e^{\sqrt{G} (2
A+B)}+\sqrt{G} H e^{\sqrt{G} (A+2 x)}+J e^{\sqrt{G} (B+2 x)}\right)}{\sqrt{G}
\left(e^{2 A \sqrt{G}}+e^{2 B \sqrt{G}}\right)} | (sqrt(G)*H*e**(sqrt(G)*(A + 2*B)) + sqrt(G)*H*e**(sqrt(G)*(A + 2*x)) - J*e**(sqrt(G)*(2*A + B)) + J*e**(sqrt(G)*(B + 2*x)))/(sqrt(G)*e**(sqrt(G)*x)*(e**(2*A*sqrt(G)) + e**(2*B*sqrt(G)))) | Symbolic-5 | ASyMOB
Differential_Equations
Q19 | Differential Equations |
185 | Solve the following differential equation:
$ y'' - 2 \cdot y' - 3 \cdot y = G \cdot \sin (x) , y(A) = H, y'(B) = J $
Assume the parameters: A,B,G,H,J are real and positive numbers. | \frac{e^{-x} \left(-e^A G \cos (A) \left(3 e^{4 B}+e^{4 x}\right)-2 e^B G \left(e^{4
A}-e^{4 x}\right) \cos (B)+6 G e^{A+4 B} \sin (A)-G e^{4 A+B} \sin (B)+30 H
e^{A+4 B}-10 J e^{4 A+B}+2 G e^{A+4 x} \sin (A)-2 G e^{4 A+x} \sin (x)+G e^{4
A+x} \cos (x)+10 H e^{A+4 x}+G e^{B+4 x} \sin (B)-6 G e^{4 B+x} \sin (x)... | (-G*e**A*(3*e**(4*B) + e**(4*x))*cos(A) + 6*G*e**(A + 4*B)*sin(A) + 2*G*e**(A + 4*x)*sin(A) - G*e**(4*A + B)*sin(B) - 2*G*e**(4*A + x)*sin(x) + G*e**(4*A + x)*cos(x) + G*e**(B + 4*x)*sin(B) - 6*G*e**(4*B + x)*sin(x) + 3*G*e**(4*B + x)*cos(x) + 30*H*e**(A + 4*B) + 10*H*e**(A + 4*x) - 10*J*e**(4*A + B) + 10*J*e**(B + 4*x... | Symbolic-5 | ASyMOB
Differential_Equations
Q20 | Differential Equations |
186 | Calculate the following infinite product. Give a finite, closed form answer. Assume the parameters: A,B are real and positive.
$ \prod_{n=1}^\infty 1-\frac{\left(\frac{A}{2 B}+1\right)^4}{B^4 (n+1)^4} $ | \frac{4 B^4 \sin \left(\frac{\pi (A+2 B)}{2 B^2}\right) \sinh \left(\frac{\pi (A+2
B)}{2 B^2}\right)}{\pi ^2 (A+2 B)^2 \left(1-\frac{(A+2 B)^4}{16 B^8}\right)} | 64*B**12*sin(pi*(A + 2*B)/(2*B**2))*sinh(pi *(A + 2*B)/(2*B**2))/(pi**2*(A + 2*B)**2*(16*B**8 - (A + 2*B)**4)) | Symbolic-2 | ASyMOB
Series
Q1 | Series |
187 | Calculate the following infinite product. Give a finite, closed form answer. Assume the parameters: A,B are real and positive.
$ \prod_{n=1}^\infty \frac{\left((A+n)^3-1\right) \left(1-\frac{1}{(B+n)^2}\right)}{(A+n)^3+1} $ | \frac{A \cdot (A+1) \cdot B}{\left(A^2+A+1\right) (B+1)} | A*B*(A + 1)/((B + 1)*(A**2 + A + 1)) | Symbolic-2 | ASyMOB
Series
Q2 | Series |
188 | Calculate the following infinite product. Give a finite, closed form answer. Assume the parameters: A,B are real and positive.
$ \prod_{n=1}^\infty 1-\frac{B^2}{(A+1)^2 n^2} $ | \frac{(A+1) \sin \left(\frac{\pi B}{A+1}\right)}{\pi B} | (A + 1)*sin(B*pi/(A + 1))/(B*pi) | Symbolic-2 | ASyMOB
Series
Q3 | Series |
189 | Calculate the following infinite product. Give a finite, closed form answer. Assume the parameters: A,B,G are real and positive.
$ \prod_{n=1}^\infty \frac{1}{A^2 \cdot n^2+\left\lfloor \frac{n}{n+B}+\left\lfloor \frac{1}{n^2 + G}\right\rfloor \right\rfloor }+1$ | \frac{A \sinh \left(\frac{\pi }{A}\right)}{\pi } | A*sinh(pi/A)/pi | Symbolic-3 | ASyMOB
Series
Q4 | Series |
190 | Calculate the following infinite product. Give a finite, closed form answer. Assume the parameters: A,J are real and positive.
$ \prod_{n=1}^\infty \left(1-\frac{J^6}{64 n^6}\right) \left(1-\frac{4}{3} \sin ^2\left(A \cdot 3^{-n}\right)\right) $ | -\frac{4 \frac{\sin( A)}{ A} \sin \left(\frac{\pi J}{2}\right) \left(\cos
\left(\frac{\pi J}{2}\right)-\cosh \left(\frac{1}{2} \sqrt{3} \pi
J\right)\right)}{\pi ^3 J^3} | -4*(cos(J*pi/2) - cosh(sqrt(3)*J*pi/2))*sin(A)*sin(J*pi/2)/(A*J**3*pi**3) | Symbolic-2 | ASyMOB
Series
Q5 | Series |
191 | Calculate the following infinite sum. Give a finite, closed form answer. Assume the parameters: A,B,G,H,J are real and positive.
$ \sum_{n=1}^\infty H \cdot \left(\frac{3}{5}\right)^n \left(G - \frac{1}{n}\right) \left(\frac{B}{A}\right)^n $ | H \left(\frac{3 B G}{5 A-3 B}+\log \left(1-\frac{3 B}{5 A}\right)\right) | H*((3*B*G + (5*A - 3*B)*log((5*A - 3*B)/(5*A)))/(5*A - 3*B)) | Symbolic-4 | ASyMOB
Series
Q6 | Series |
192 | Calculate the following infinite sum. Give a finite, closed form answer. Assume the parameters: A,B are real and positive.
$ \sum_{n=1}^\infty \frac{(-A)^n}{B \cdot (2 n+5)} $ | \frac{\frac{15 \tan^{-1}\left(\sqrt{A}\right)}{A^{5/2}}-\frac{15}{A^2}+\frac{5}{A}-3}{15 B} | -1/(5*B) + 1/(3*A*B) - 1/(A**2*B) + atan(sqrt(A))/(A**(5/2)*B) | Symbolic-2 | ASyMOB
Series
Q7 | Series |
193 | Calculate the following infinite sum. Give a finite, closed form answer. Assume the parameters: A,B,G,H are real and positive.
$ \sum_{n=1}^\infty 3^{A + H \cdot n} \cdot 2^{B - 3 \cdot G \cdot n} $ | \frac{2^B 3^{A+H}}{8^G-3^H} | -2**B*3**(A + H)/(3**H - 8**G) | Symbolic-4 | ASyMOB
Series
Q8 | Series |
194 | Calculate the following infinite sum. Give a finite, closed form answer. Assume the parameters: A,B,G are real and positive.
$ \sum_{n=1}^\infty \frac{B \cdot A^{G \cdot n}}{n!} $ | B \left(e^{A^G}-1\right) | B*(e**(A**G) - 1) | Symbolic-3 | ASyMOB
Series
Q9 | Series |
195 | Calculate the following infinite sum. Give a finite, closed form answer. Assume the parameters: A,B are real and positive.
$ \sum_{n=1}^\infty \frac{A}{-B + n^2 - n} $ | \frac{\pi A \tan \left(\frac{1}{2} \pi \sqrt{4 B+1}\right)}{\sqrt{4 B+1}} | A*pi*tan(pi*sqrt(4*B + 1)/2)/sqrt(4*B + 1) | Symbolic-2 | ASyMOB
Series
Q10 | Series |
196 | Calculate the following infinite sum. Give a finite, closed form answer. Assume the parameters: A,B,G,H,J are real and positive.
$ \sum_{n=1}^\infty \frac{A \cdot n^3 + B \cdot n^2 + H \cdot n + J}{G \cdot n \cdot (n+1)^2 \cdot (n+2)^2} $ | \frac{2 \left(5 \pi ^2-48\right) A-6 \left(\pi ^2-10\right) B-39 H+4 \pi ^2 H+30 J-3
\pi ^2 J}{12 G} | (2*A*(5*pi**2 - 48) - 6*B*(pi**2 - 10) + 4*H*pi**2 - 39*H - 3*J*pi**2 + 30*J)/(12*G) | Symbolic-5 | ASyMOB
Series
Q11 | Series |
197 | Calculate the following infinite sum. Give a finite, closed form answer. Assume the parameters: A,B,G are real and positive.
$ \sum_{n=1}^\infty \frac{A \cdot n+B}{G \cdot (n+4) \cdot (n+2) \cdot (n+3)} $ | \frac{4 A+B}{24 G} | (4*A + B)/(24*G) | Symbolic-3 | ASyMOB
Series
Q12 | Series |
198 | Calculate the following infinite sum. Give a finite, closed form answer. Assume the parameters: A,B,G are real and positive.
$ \sum_{n=1}^\infty \frac{A \cdot (-1)^n \cdot n + B}{G \cdot (n+1) \cdot (n+2) \cdot (n+3)} $ | \frac{A (33-48 \log (2))+B}{12 G} | (B + A*(33 - 48*log(2)))/(12*G) | Symbolic-3 | ASyMOB
Series
Q13 | Series |
199 | Calculate the following infinite sum. Give a finite, closed form answer. Assume the parameters: A,B,G,H,J are real and positive.
$ \sum_{n=1}^\infty \frac{(-2)^{-n} \cdot {J}^{-n} \cdot \left(A \cdot (-1)^n + B \cdot n^2 + H \cdot n \right)}{G} $ | \frac{\frac{A}{2 J - 1}-\frac{2 J \cdot (B \cdot (2 J - 1) + 2 H J + H)}{(2 J+1)^3}}{G} | (A*(2*J + 1)**3 - 2*J*(2*J - 1)*(B*(2*J - 1) + 2*H*J + H))/(G*(2*J - 1)*(2*J + 1)**3) | Symbolic-5 | ASyMOB
Series
Q14 | Series |
200 | Calculate the following infinite sum. Give a finite, closed form answer. Assume the parameters: A,B,G,H,J are real and positive.
$ \sum_{n=1}^\infty \frac{A \cdot (-1)^{B + n} + H}{G \cdot \left(J^2 + n^2\right)} $ | \frac{\pi \cdot A \cdot (-1)^{B} \cdot J \cdot \frac{1}{\sinh(\pi \cdot J)}+A \cdot \left(-(-1)^{B}\right)+\pi H \cdot J \cdot \coth (\pi \cdot J) - H}{2 \cdot G \cdot J^2} | ((-1)**B*A*J*pi/sinh(J*pi) + (-1)**(B + 1)*A + H*J*pi*coth(J*pi) - H)/(2*G*J**2) | Symbolic-5 | ASyMOB
Series
Q15 | Series |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.