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 |
|---|---|---|---|---|---|---|
1 | Compute the first 5 nonzero terms of the Maclaurin series of $ e^{\sin(x)} $ | 1+x+\frac{x^2}{2}-\frac{x^4}{8}-\frac{x^5}{15}+\cdots | -x**5/15 - x**4/8 + x**2/2 + x + 1 | Original | U-Math
sequences_series
1ccc052c-9604-4459-a752-98ebdf3e0764 | Series |
2 | Find the radius of convergence of the series:
$ \sum_{n=1}^\infty \left(\frac{ \left((2 \cdot n)!\right) \cdot x^n }{ n^{2 \cdot n} }\right) $ | \frac{e^2}{4} | E**2 / 4 | Original | U-Math
sequences_series
ca5ffe7c-f495-43dc-a653-de477cabc185 | Series |
3 | Compute the first 6 nonzero terms of the Maclaurin series of $f(x) = \sin(x) \cdot \cos\left(\frac{ \pi }{ 4 }\right) + \cos(x) \cdot \sin\left(\frac{ \pi }{ 4 }\right)$ | \frac{1}{34560\cdot\sqrt{2}}\cdot\left(288\cdot x^5+1440\cdot x^4-5760\cdot x^3-17280\cdot x^2+34560\cdot x+34560\right) | sqrt(2)*(x**5 + 5*x**4 - 20*x**3 - 60*x**2 + 120*x + 120)/240 | Original | U-Math
sequences_series
f89bd354-18c9-4f31-b91f-cf6421e24921 | Series |
4 | Compute the first 4 nonzero terms of the Maclaurin series of $f(x) = e^x \cdot \cos(x)$ | 1+x-\frac{x^3}{3}-\frac{x^4}{6} | -x**4/6 - x**3/3 + x + 1 | Original | U-Math
sequences_series
d1fe21df-ee7f-40c2-9655-6bd6a7a23ff1 | Series |
5 | Compute $\lim_{x \to 0}\left(\frac{ 2 \cdot \cos(x)+4 }{ 5 \cdot x^3 \cdot \sin(x) }-\frac{ 6 }{ 5 \cdot x^4 }\right)$ | \frac{1}{150} | 1/150 | Original | U-Math
sequences_series
068e40ce-9108-4ef8-8ee5-0d1471ebbe43 | Limits |
6 | Evaluate $\lim_{x \to 0^{+}}\left(\left(\frac{ \tan\left(\frac{ x }{ 2 }\right) }{ \frac{ x }{ 2 } }\right)^{\frac{ 3 }{ x^2 }}\right)$ | $e^{\frac{1}{4}}$ | exp(1/4) | Original | U-Math
differential_calc
363dd580-f1fc-4867-a6ef-db2a03139745 | Limits |
7 | Evaluate
$ \lim_{x \to 5} \left( \frac{ 3 \cdot x }{ x-5 }-\frac{ 3 }{ \ln\left(\frac{ x }{ 5 }\right) } \right) $ | \frac{3}{2} | 3/2 | Original | U-Math
differential_calc
2d799998-115a-489b-a48b-57090954303e | Limits |
8 | Evaluate
$ \lim_{x \to \infty} \left(x - x^2 \cdot \ln\left(1 + \frac{ 1 }{ x }\right)\right) $ | \frac{1}{2} | 1/2 | Original | U-Math
differential_calc
efdc4110-cf56-4f37-bf54-40fdd5d58145 | Limits |
9 | Evaluate
$ \lim_{x \to 0^{+}} \left( \left( \frac{ \tan(2 \cdot x) }{ 2 \cdot x } \right)^{\frac{ 1 }{ 3 \cdot x^2 }} \right) $ | e^{\frac{4}{9}} | e**(4/9) | Original | U-Math
differential_calc
99a2304d-5d8e-4245-90da-a80651ca15d8 | Limits |
10 | Evaluate
$ \lim_{x \to 0}\left( \left| \frac{ -\sin(x) }{ x } \right| \right)^{\frac{ 1 }{ 4 \cdot x^2 }} $ | e^{\frac{-1}{24}} | e**(-1/24) | Original | U-Math
differential_calc
84c6a419-c103-41d5-aad5-dd8e690c6e88 | Limits |
11 | Integrate
$ \int \sin(x)^4 \cdot \cos(x)^6 dx $ | $C+\frac{1}{320}\cdot\left(\sin(2\cdot x)\right)^5+\frac{1}{128}\cdot\left(\frac{3\cdot x}{2}-\frac{\sin(4\cdot x)}{2}+\frac{\sin(8\cdot x)}{16}\right)$ | 3*x/256 + sin(2*x)**5/320 - sin(4*x)/256 + sin(8*x)/2048 | Original | U-Math
integral_calc
0c0ba3db-1470-4c36-975c-91ff5f51986f | Integrals |
12 | Calculate the integral:
$
\int \frac{ \sqrt[5]{x}+\sqrt[5]{x^4}+x \cdot \sqrt[5]{x} }{ x \cdot \left(1+\sqrt[5]{x^2}\right) } dx
$ | C+5\cdot\arctan\left(\sqrt[5]{x}\right)+\frac{5}{4}\cdot\sqrt[5]{x}^4 | C + 5*x**(4/5)/4 + 5*atan(x**(1/5)) | Original | U-Math
integral_calc
126c4165-b3d5-4470-8412-08e79d9821cf | Integrals |
13 | Solve the integral:
$
\int \frac{ 1 }{ \sin(x)^7 \cdot \cos(x) } dx
$ | C+\ln\left(\left|\tan(x)\right|\right)-\frac{3}{2\cdot\left(\tan(x)\right)^2}-\frac{3}{4\cdot\left(\tan(x)\right)^4}-\frac{1}{6\cdot\left(\tan(x)\right)^6} | C + log(Abs(tan(x))) - 3 / (2 * tan(x)**2) - 3 / (4 * tan(x)**4) - 1 / (6 * tan(x)**6) | Original | U-Math
integral_calc
00f6affb-905a-4109-a78e-2dde7a0b83accf | Integrals |
14 | Compute the integral:
$
-2 \cdot \int x^{-4} \cdot \left(4+x^2\right)^{\frac{ 1 }{ 2 }} dx
$ | C+\frac{1}{6}\cdot\left(\frac{4}{x^2}+1\right)\cdot\sqrt{\frac{4}{x^2}+1} | (C*x**2 + sqrt((x**2 + 4)/x**2)*(x**2 + 4)/6)/x**2 | Original | U-Math
integral_calc
05ea9929-8cbb-432b-bbbb-ec1e74c9f401 | Integrals |
15 | Solve the integral:
$
\int \left(\frac{ x+4 }{ x-4 } \right)^{\frac{ 3 }{ 2 }} dx
$ | C+\sqrt{\frac{x+4}{x-4}}\cdot(x-20)-12\cdot\ln\left(\left|\frac{\sqrt{x-4}-\sqrt{x+4}}{\sqrt{x-4}+\sqrt{x+4}}\right|\right) | C + sqrt((x + 4)/(x - 4)) * (x - 20) - 12 * log(Abs((sqrt(x - 4) - sqrt(x + 4)) / (sqrt(x - 4) + sqrt(x + 4)))) | Original | U-Math
integral_calc
08c72d46-1abd-49e1-9c9c-ce509902be6e | Integrals |
16 | Compute the integral: $ \int \frac{ -1 }{ x^2 \cdot \left(3+x^3\right)^{\frac{ 5 }{ 3 }} } dx $ | \frac{1}{9}\cdot\sqrt[3]{1+\frac{3}{x^3}}+\frac{1}{18\cdot\left(1+\frac{3}{x^3}\right)^{\frac{2}{3}}} | (x**3 + 2)/(6*x**3*(1 + 3/x**3)**(2/3)) | Original | U-Math
integral_calc
4c1292e1-d4b3-4acf-afaf-eaac62f2662d | Integrals |
17 | Compute the integral:
$ \int \frac{ 4 \cdot x+\sqrt{4 \cdot x-5} }{ 5 \cdot \sqrt[4]{4 \cdot x-5}+\sqrt[4]{(4 \cdot x-5)^3} } dx $ | C+25\cdot\sqrt[4]{4\cdot x-5}+\frac{1}{5}\cdot\sqrt[4]{4\cdot x-5}^5-\frac{4}{3}\cdot\sqrt[4]{4\cdot x-5}^3-\frac{125}{\sqrt{5}}\cdot\arctan\left(\frac{1}{\sqrt{5}}\cdot\sqrt[4]{4\cdot x-5}\right) | C + (4*x - 5)**(5/4)/5 - 4*(4*x - 5)**(3/4)/3 + 25*(4*x - 5)**(1/4) - 25*sqrt(5)*atan(sqrt(5)*(4*x - 5)**(1/4)/5) | Original | U-Math
integral_calc
147944c5-b782-48c5-a664-d66deb92d9a7 | Integrals |
18 | Solve the integral:
$
\int \frac{ 3 }{ \sin(2 \cdot x)^7 \cdot \cos(-2 \cdot x) } dx
$ | C+\frac{3}{2}\cdot\left(\ln\left(\left|\tan(2\cdot x)\right|\right)-\frac{3}{2\cdot\left(\tan(2\cdot x)\right)^2}-\frac{3}{4\cdot\left(\tan(2\cdot x)\right)^4}-\frac{1}{6\cdot\left(\tan(2\cdot x)\right)^6}\right) | C + (3/2) * (log(Abs(tan(2*x))) - 3/(2 * tan(2*x)**2) - 3/(4 * tan(2*x)**4) - 1/(6 * tan(2*x)**6)
) | Original | U-Math
integral_calc
1db212f0-2fac-410d-969d-fe3b5b55d076 | Integrals |
19 | Solve the integral:
$
\int \frac{ 1 }{ (\sin(8 \cdot x))^5 } dx
$ | C+\frac{1}{128}\cdot\left(2\cdot\left(\tan(4\cdot x)\right)^2+6\cdot\ln\left(\left|\tan(4\cdot x)\right|\right)+\frac{1}{4}\cdot\left(\tan(4\cdot x)\right)^4-\frac{2}{\left(\tan(4\cdot x)\right)^2}-\frac{1}{4\cdot\left(\tan(4\cdot x)\right)^4}\right) | C + Rational(1, 128) * (2 * tan(4 * x)**2 + 6 * log(Abs(tan(4 * x))) + Rational(1, 4) * tan(4 * x)**4 - 2 / tan(4 * x)**2 - 1 / (4 * tan(4 * x)**4)) | Original | U-Math
integral_calc
275f7ceb-f331-4a3f-96ec-346e6d81b32a | Integrals |
20 | Evaluate the integral:
$ \int \left(x^3 + 3\right) \cdot \cos(2 \cdot x) dx $ | \frac{1}{256}\cdot\left(384\cdot\sin(2\cdot x)+128\cdot x^3\cdot\sin(2\cdot x)+192\cdot x^2\cdot\cos(2\cdot x)-96\cdot\cos(2\cdot x)-256\cdot C-192\cdot x\cdot\sin(2\cdot x)\right) | -C + x**3*sin(2*x)/2 + 3*x**2*cos(2*x)/4 - 3*x*sin(2*x)/4 + 3*sin(2*x)/2 - 3*cos(2*x)/8 | Original | U-Math
integral_calc
47a11349-0386-4969-9263-d3cdfcc98cb9 | Integrals |
21 | Use factoring to calculate the following limit.
$ \lim_{x \rightarrow K} \frac {{x}^4-K^4} {{x}^5-K^5} $ | \frac{4}{5 K} | 4/(5*K) | Original | UGMathBench
Calculus_-_single_variable_0016 | Limits |
22 | Find the limit. $ \lim_{x \to 0} \frac{1-\cos\!\left(10x\right)}{\cos^{2}\!\left(6x\right)-1}$ | \frac{-25}{18} | -25/18 | Original | UGMathBench
Calculus_-_single_variable_0022 | Limits |
23 | Evaluate the limit. $ \lim_{x\to 1} \dfrac{x^2+11x-12}{\ln x}=$ | 13 | 13 | Original | UGMathBench
Calculus_-_single_variable_0508 | Limits |
24 | Evaluate the limit below, given that $f(t)=\left(\frac{4^t+6^t}{4}\right)^{1/t}$. $\lim\limits_{t\to+\infty} f(t)$ | 6 | 6 | Original | UGMathBench
Calculus_-_single_variable_0512 | Limits |
25 | Calculate the integral. $\int_{2}^{\infty} 3x^{2}e^{-x^{3}} dx$ | \frac{1}{e^{8}} | e**(-8) | Original | UGMathBench
Calculus_-_single_variable_0592 | Integrals |
26 | Evaluate the indefinite integral. $\int \tan^{3}\!\left(x\right)\sec^{9}\!\left(x\right) dx$ | \frac{\sec^{11}{\left(x \right)}}{11} - \frac{\sec^{9}{\left(x \right)}}{9} | sec(x)**11/11 - sec(x)**9/9 | Original | UGMathBench
Calculus_-_single_variable_0604 | Integrals |
27 | Evaluate the indefinite integral.
$\int 208 \cos^4(16x) dx$ | 78 x + \frac{13 \sin{\left(16 x \right)} \cos^{3}{\left(16 x \right)}}{4} + \frac{39 \sin{\left(16 x \right)} \cos{\left(16 x \right)}}{8} | 78*x + 13*sin(16*x)*cos(16*x)**3/4 + 39*sin(16*x)*cos(16*x)/8 | Original | UGMathBench
Calculus_-_single_variable_0606 | Integrals |
28 | Evaluate the integral.
$ \int \frac{10x^2-48x-38}{x^3-5x^2-8x+48} dx $ | \frac{2 \left(\left(x - 4\right) \left(3 \log{\left(\left|{x - 4}\right| \right)} + 2 \log{\left(\left|{x + 3}\right| \right)}\right) + 5\right)}{x - 4} | 2*((x - 4)*(3*log(Abs(x - 4)) + 2*log(Abs(x + 3))) + 5)/(x - 4) | Original | UGMathBench
Calculus_-_single_variable_0612 | Integrals |
29 | Evaluate the integral. $ \int e^{x}\sqrt{64-e^{2x}} dx$ | \frac{e^{x} \sqrt{64 - e^{2 x}}}{2} + 32 \operatorname{asin}{\left(\frac{e^{x}}{8} \right)} | e**x*sqrt(64 - e**(2*x))/2 + 32*asin(e**x/8) | Original | UGMathBench
Calculus_-_single_variable_0624 | Integrals |
30 | Evaluate $\lim_{x \to 0} \frac{e^{-3x^3}-1+3x^3-\frac{9}{2}x^6}{12x^9}$ | \frac{-3}{8} | -3/8 | Original | UGMathBench
Calculus_-_single_variable_0939 | Limits |
31 | Solve the following first-order differential equation:
$ \frac{dy}{dx} + 2y = e^{-x}, \quad y(0) = 1. $ | e^{-x} | e**(-x) | Original | MathOdyssey
Problem 340 from Differential Equations - College Math | Differential Equations |
32 | Consider the differential equation $\frac{dy}{dx} = xy$. Find the value of $y(\sqrt{2})$ given that $y(0) = 2$. | 2e | 2*e | Original | MathOdyssey
Problem 339 from Differential Equations - College Math | Differential Equations |
33 | Evaluate the following limit:
$
\lim_{n \to \infty} \left(\sqrt{n^2+2n-1}-\sqrt{n^2+3}\right).
$ | 1 | 1 | Original | MathOdyssey
Problem 315 from Calculus and Analysis - College Math | Limits |
34 | Evaluate $\lim\limits_{x\to 4}\frac{x-4}{\sqrt{x}-2}$. | 4 | 4 | Original | MathOdyssey
Problem 317 from Calculus and Analysis - College Math | Limits |
35 | Evaluate $ \int_0^4(2x-\sqrt{16-x^2})dx$. | 16 - 4 \pi | 16 - 4*pi | Original | MathOdyssey
Problem 325 from Calculus and Analysis - College Math | Integrals |
36 | Evaluate the series $\sum\limits_{n=1}^\infty\frac{1}{(n+1)(n+3)}$. | \frac{5}{12} | 5/12 | Original | MathOdyssey
Problem 326 from Calculus and Analysis - College Math | Series |
37 | Evaluate the limit $\lim\limits_{x\to 0}\frac{(1+x)^{\frac{1}{x}}-e}{x}$. | -\frac{ e}{2} | -e/2 | Original | MathOdyssey
Problem 327 from Calculus and Analysis - College Math | Limits |
38 | Evaluate the series $\sum\limits_{n=0}^\infty \frac{1}{2n+1}\left(\frac12\right)^{2n+1}$. | \ln\sqrt{3} | log(3)/2 | Original | MathOdyssey
Problem 328 from Calculus and Analysis - College Math | Series |
39 | Evaluate the limit $\lim\limits_{n\to\infty}\sum\limits_{k=0}^{n-1}\frac{1}{\sqrt{n^2-k^2}}$. | \frac{\pi}{2} | pi/2 | Original | MathOdyssey
Problem 329 from Calculus and Analysis - College Math | Limits |
40 | Evaluate the iterated integral $\displaystyle{\int_0^1dy\int_y^1(e^{-x^2}+e^x)dx}$. | \frac{3}{2}-\frac12 e^{-1} | (3*e - 1)/(2*e) | Original | MathOdyssey
Problem 336 from Calculus and Analysis - College Math | Integrals |
41 | What is the integral of $ 2x - x^7atan(3) $ | x^2-\frac{1}{8} x^8 \tan ^{-1}(3) | -x**8*atan(3)/8 + x**2 | Original | GHOSTS
Symbolic Integration
Q97 | Integrals |
42 | What is the integral of $ 1 + x + x^3*cosh(2) $ | \frac{1}{4} x^4 \cosh (2)+\frac{x^2}{2}+x | x**4*cosh(2)/4 + x**2/2 + x | Original | GHOSTS
Symbolic Integration
Q98 | Integrals |
43 | What is the integral of $ 12 + 6cosh(x) $ | 12 x + 6 \sinh{\left(x \right)} | 12*x + 6*sinh(x) | Original | GHOSTS
Symbolic Integration
Q90 | Integrals |
44 | What is the integral of 4x^7 + sin(1 + x) | \frac{x^8}{2} - \cos(1+x) | x**8/2 - cos(x + 1) | Original | GHOSTS
Symbolic Integration
Q14 | Integrals |
45 | What is the integral of 2x + 2x^2 + x*[(x + x*e^x)^(-1)] | \frac{2 x^3}{3}+x^2-2 \tanh ^{-1}\left(2 e^x+1\right) | 2*x**3/3 + x**2 + x - log(exp(x) + 1) | Original | GHOSTS
Symbolic Integration
Q7 | Integrals |
46 | What is the integral of -x + cos[ln(sin(3))] * ln(3x) | -\frac{1}{2} x (x-2 \log (3 x) \cos (\log (\sin (3)))+2 \cos (\log (\sin (3)))) | -1*x*((x - 2*log(3*x, E)*cos(log(sin(3), E))) + 2*cos(log(sin(3), E)))/2 | Original | GHOSTS
Symbolic Integration
Q15 | Integrals |
47 | What is the integral of 3x - 4*[cos(x+3)]*x^2 | \frac{3 x^2}{2}-4 \left(x^2-2\right) \sin (x+3)-8 x \cos (x+3) | -8*x*cos(x + 3) + ((3*x**2)/2 - 4*(x**2 - 2)*sin(x + 3)) | Original | GHOSTS
Symbolic Integration
Q18 | Integrals |
48 | What is the integral of -3 + atan(x) + ln(tanh(3)) | x \arctan(x) - \frac{1}{2} \ln(1 + x^2) + x \ln(\tanh(3)) - 3x + C | x*atan(x) - 3*x + x*log(tanh(3)) - log(x**2 + 1)/2 | Original | GHOSTS
Symbolic Integration
Q20 | Integrals |
49 | What is the integral of e^{x \left(x + 4\right)^{2}} \left(x + 4\right) \left(3 x + 4\right) | e^{x (x+4)^2} | e**(x*(x + 4)**2) | Original | GHOSTS
Symbolic Integration
Q22 | Integrals |
50 | What is the integral of -e^{3x} * sin(e^{3x}) | \frac{1}{3} \cos \left(e^{3 x}\right) | cos(e**(3*x))/3 | Original | GHOSTS
Symbolic Integration
Q29 | Integrals |
51 | If $\log _{2} x-2 \log _{2} y=2$, determine $y$, as a function of $x$ | \frac{1}{2} \sqrt{x} | sqrt(x)/2 | Original | OlympiadBench
oe_to_maths_en_comp
2498 | Differential Equations |
52 | If $f(x)=2 x+1$ and $g(f(x))=4 x^{2}+1$, determine an expression for $g(x)$. | x^2-2 x+2 | x**2 - 2*x + 2 | Original | OlympicArena
Math_1381 | Series |
53 | Solve the following integral $\int_0^{\frac{\pi}{2}} \frac{x \sin(2x)}{1 + \cos^2(2x)} dx$ | Pi^2 / 16 | pi**2 / 16 | Original | OBMU 2019 - Q21 | Integrals |
54 | Solve the following integral:
$\int_{1}^{2} \frac{e^x(x - 1)}{x(x + e^x)} dx$ | \ln\left( \frac{2 + e^2}{2 + 2e} \right) | log((E**2 + 2)/(2*E + 2), E) | Original | OBMU 2019 - Q18 | Integrals |
55 | Solve the following integral:
$\int_{0}^{\pi} \log(\sin(x)) dx$ | -\pi \log (2) | -pi*log(2, E) | Original | OBMU 2019 - Q22 | Integrals |
56 | Evaluate the following hypergeometric function. Return a closed-form symbolic answer.
$ {}_2F_1\left( \begin{array}{c} 1,1\ \\ 2 \end{array}; -1 \right) $ | \log (2) | log(2, E) | Original | ASyMOB
Hypergeometrics
Q1 | Hypergeometrics |
57 | Evaluate the following hypergeometric function. Return a closed-form symbolic answer.
$ {}_2F_1\left( \begin{array}{c} 1,1 \\ 3 \end{array}; -2 \right) $ | \frac{3 \log (3)}{2}-1 | -1 + (3*log(3, E))/2 | Original | ASyMOB
Hypergeometrics
Q2 | Hypergeometrics |
58 | Evaluate the following hypergeometric function. Return a closed-form symbolic answer.
$ {}_3F_2\left( \begin{array}{c} 1,1,1 \\ 2,2 \end{array}; -1 \right) $ | \frac{\pi ^2}{12} | pi**2/12 | Original | ASyMOB
Hypergeometrics
Q3 | Hypergeometrics |
59 | Evaluate the following hypergeometric function. Return a closed-form symbolic answer.
$ {}_3F_2\left( \begin{array}{c} -1,-1,-1 \\ -1,-1 \end{array}; x \right) $ | 1-x | 1-x | Original | ASyMOB
Hypergeometrics
Q4 | Hypergeometrics |
60 | Solve the following integral. Return a closed-form symbolic answer.
\int \frac{ 1 }{ 1 + x^3 } dx | -\frac{1}{6} \log \left(x^2-x+1\right)+\frac{1}{3} \log (x+1)+\frac{\tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)}{\sqrt{3}} | (log(x + 1, E)/3 - 1*log((x**2 - x) + 1, E)/6) + atan((2*x - 1)/(sqrt(3)))/(sqrt(3)) | Original | ASyMOB
Hypergeometrics
Q5 | Hypergeometrics |
61 | Solve the following integral.
\int \frac{(4 + (4 - 1)x^1)x^{2-1}}{2(1 + x^1 + x^{4})\sqrt{1 + x^1}} dx | \tan ^{-1}\left(\frac{x^2}{\sqrt{x+1}}\right) | atan(x**2/sqrt(x + 1)) | Original | ASyMOB
Hypergeometrics
Q6 | Hypergeometrics |
62 | Evaluate the following hypergeometric function. Return a closed-form symbolic answer.
$ {}_2F_1\left( \begin{array}{c} 1, 1 \\ 1 \end{array}; -1 \right) $ | \frac{1}{2} | 1/2 | Original | ASyMOB
Hypergeometrics
Q7 | Hypergeometrics |
63 | Evaluate the following hypergeometric function.Return a closed-form symbolic answer.
$ {}_1F_1\left( \begin{array}{c} 1 \\ 1 \end{array}; 1 \right) $ | e | E | Original | ASyMOB
Hypergeometrics
Q8 | Hypergeometrics |
64 | Evaluate the following hypergeometric function. Return a closed-form symbolic answer.
$ {}_2F_1\left( \begin{array}{c} 1, -2 \\ 2+1 \end{array}; 1 \right) $ | \frac{1}{2} | 1/2 | Original | ASyMOB
Hypergeometrics
Q9 | Hypergeometrics |
65 | Evaluate the following hypergeometric function. Return a closed-form symbolic answer.
$ {}_3F_1\left( \begin{array}{c} (1 + 1), -2, (1 + 3) \\ (2 + 1) \end{array}; \frac{1}{2} \right) $ | \frac{5}{6} | 5/6 | Original | ASyMOB
Hypergeometrics
Q10 | Hypergeometrics |
66 | Solve the following differential equation:
$ x \cdot y' + y = x \cdot \sin(x), y(\pi) = 1 $ | -\cos(x) + \frac{\sin(x)}{x} | -cos(x) + sin(x)/x | Original | ASyMOB
Differential_Equations
Q1 | Differential Equations |
67 | Solve the following differential equation:
$ y' = e^{ x } \cdot y , y(1) = 1 $ | e^{e^x-e} | e**(-e + e**x) | Original | ASyMOB
Differential_Equations
Q2 | Differential Equations |
68 | Solve the following differential equation:
$ y' = 2 \cdot x \cdot y^2 - y , y(1) = 1 $ | \frac{e}{2 e x-3 e^x+2 e} | e/(2*e*x + 2*e - 3*e**x) | Original | ASyMOB
Differential_Equations
Q3 | Differential Equations |
69 | Solve the following differential equation:
$ y' = x y^2 - y , y(1) = 1 $ | \frac{e}{e x-e^x+e} | e/(e*x + e - e**x) | Original | ASyMOB
Differential_Equations
Q4 | Differential Equations |
70 | Solve the following differential equation:
$ y' = \frac{y^2 + 2 \cdot x \cdot y }{x^2} , y(1) = 1 $ | -\frac{x^2}{x-2} | -x**2/(x - 2) | Original | ASyMOB
Differential_Equations
Q5 | Differential Equations |
71 | Solve the following differential equation:
$ y' = e^{-x} - 2 \cdot y , y(1) = 1 $ | e^{-2 x} \left(e^x-e+e^2\right) | (e**2 - e + e**x)/e**(2*x) | Original | ASyMOB
Differential_Equations
Q6 | Differential Equations |
72 | Solve the following differential equation:
$ y' = 3 \cdot x^2 \cdot \left( y^2 + 1 \right) , y(1) = 1 $ | \tan \left(x^3+\frac{1}{4} (\pi -4)\right) | tan(pi/4 + x**3 - 1) | Original | ASyMOB
Differential_Equations
Q7 | Differential Equations |
73 | Solve the following differential equation:
$ y' = \frac{2 (y + x)}{x} , y(1) = 1 $ | 3 x^2-2 x | x*(3*x - 2) | Original | ASyMOB
Differential_Equations
Q8 | Differential Equations |
74 | Solve the following differential equation:
$ y' = \frac{ (y + x)}{x} , y(1) = 1 $ | x+x \log (x) | x*(log(x) + 1) | Original | ASyMOB
Differential_Equations
Q9 | Differential Equations |
75 | Solve the following differential equation:
$ y' = x , y(1) = 1 $ | \frac{1}{2} \left(x^2+1\right) | x**2/2 + 1/2 | Original | ASyMOB
Differential_Equations
Q10 | Differential Equations |
76 | Solve the following differential equation:
$ y' = x - 2 \cdot x \cdot y , y(1) = 1 $ | \frac{1}{2} e^{-x^2} \left(e^{x^2}+e\right) | e/(2*e**(x**2)) + 1/2 | Original | ASyMOB
Differential_Equations
Q11 | Differential Equations |
77 | Solve the following differential equation:
$ y' = -y + 2 \cdot \sin (x) + 5 \cdot \sin (2 \cdot x) , y(0) = 0 $ | -e^{-x} \left(-e^x \sin (x)-e^x \sin (2 x)+e^x \cos (x)+2 e^x \cos (2 x)-3\right) | sin(2*x) - 2*cos(2*x) - sqrt(2)*cos(x + pi/4) + 3/e**x | Original | ASyMOB
Differential_Equations
Q12 | Differential Equations |
78 | Solve the following differential equation:
$ y' = \tan (y) , y(1) = 1 $ | \sin ^{-1}\left(e^{x-1} \sin (1)\right) | asin(e**(x - 1)*sin(1)) | Original | ASyMOB
Differential_Equations
Q13 | Differential Equations |
79 | Solve the following differential equation:
$ y' = \sin ^2(y) + 2 \cos ^2(y) , y(1) = 1 $ | -\tan ^{-1}\left(\sqrt{2} \tan \left(-\sqrt{2} x+\sqrt{2}-\tan
^{-1}\left(\frac{\tan (1)}{\sqrt{2}}\right)\right)\right) | atan(sqrt(2)*tan(sqrt(2)*x - sqrt(2) + atan(sqrt(2)*tan(1)/2))) | Original | ASyMOB
Differential_Equations
Q14 | Differential Equations |
80 | Solve the following differential equation:
$ y' = 2 \cos ^2(y) - \sin ^2(y) , y(0) = 0 $ | \tan ^{-1}\left(\sqrt{2} \tanh \left(\sqrt{2} x\right)\right) | atan(sqrt(2)*tanh(sqrt(2)*x)) | Original | ASyMOB
Differential_Equations
Q15 | Differential Equations |
81 | Solve the following differential equation:
$ y'' + y' + y + x = 0 , y(0) = 1 , y'(0) = 1 $ | -\frac{1}{3} e^{-x/2} \left(3 e^{x/2} x-3 e^{x/2}-4 \sqrt{3} \sin
\left(\frac{\sqrt{3} x}{2}\right)\right) | -x + 1 + 4*sqrt(3)*sin(sqrt(3)*x/2)/(3*e**(x/2)) | Original | ASyMOB
Differential_Equations
Q16 | Differential Equations |
82 | Solve the following differential equation:
$ y'' + y' - 6 \cdot y = 0 , y(0) = 0 , y'(0) = 1 $ | \frac{1}{5} e^{-3 x} \left(e^{5 x}-1\right) | (e**(5*x) - 1)/(5*e**(3*x)) | Original | ASyMOB
Differential_Equations
Q17 | Differential Equations |
83 | Solve the following differential equation:
$ y'' + y = 0 , y(1) = 1 , y'(0) = 1 $ | \sin (x)+\cos (x) (\sec (1)-\tan (1)) | sin(x) - (-sec(1) + tan(1))*cos(x) | Original | ASyMOB
Differential_Equations
Q18 | Differential Equations |
84 | Solve the following differential equation:
$ y'' - y = 0 , y(1) = 1 , y'(1) = 1 $ | e^{x-1} | e**(x - 1) | Original | ASyMOB
Differential_Equations
Q19 | Differential Equations |
85 | Solve the following differential equation:
$ y'' - 2 \cdot y' - 3 \cdot y = \sin (x) , y(1) = 1 , y'(1) = 1 $ | \frac{1}{40} e^{-x-3} \left(20 e^{4 x}-8 e^{x+3} \sin (x)+3 e^{4 x} \sin (1)+4
e^{x+3} \cos (x)+e^{4 x} \cos (1)+20 e^4+5 e^4 \sin (1)-5 e^4 \cos (1)\right) | e**(-x - 3)*(-5*sqrt(2)*e**4*cos(pi/4 + 1) + 20*e**4 + e**(4*x)*cos(1) + 3*e**(4*x)*sin(1) + 20*e**(4*x) - 8*e**(x + 3)*sin(x) + 4*e**(x + 3)*cos(x))/40 | Original | ASyMOB
Differential_Equations
Q20 | Differential Equations |
86 | Calculate the following infinite product. Give a finite, closed form answer.
$ \prod_{n=1}^\infty 1-\frac{81}{16 (n+1)^4} $ | \frac{64 \sinh \left(\frac{3 \pi }{2}\right)}{585 \pi ^2} | 64*sinh(3*pi/2)/(585*pi**2) | Original | ASyMOB
Series
Q1 | Series |
87 | Calculate the following infinite product. Give a finite, closed form answer.
$ \prod_{n=1}^\infty \frac{\left(1-\frac{1}{(n+1)^2}\right) \left((n+1)^3-1\right)}{(n+1)^3+1} $ | \frac{1}{3} | 1/3 | Original | ASyMOB
Series
Q2 | Series |
88 | Calculate the following infinite product. Give a finite, closed form answer.
$ \prod_{n=1}^\infty 1-\frac{1}{4 n^2} $ | \frac{2}{\pi } | 2/pi | Original | ASyMOB
Series
Q3 | Series |
89 | Calculate the following infinite product. Give a finite, closed form answer.
$ \prod_{n=1}^\infty \frac{1}{\left\lfloor \frac{n}{n+1}+\left\lfloor \frac{1}{n^2+1}\right\rfloor
\right\rfloor +n^2}+1 $ | \frac{\sinh (\pi )}{\pi } | sinh(pi)/pi | Original | ASyMOB
Series
Q4 | Series |
90 | Calculate the following infinite product. Give a finite, closed form answer.
$ \prod_{n=1}^\infty \left(1-\frac{1}{64 \cdot n^6}\right) \left(1-\frac{4}{3} \sin^2\left(3^{-n}\right)\right) $ | \frac{4 \frac{\sin(1)}{ 1} \cosh \left(\frac{\sqrt{3} \pi }{2}\right)}{\pi ^3} | 4*sin(1)*cosh(sqrt(3)*pi/2)/pi**3 | Original | ASyMOB
Series
Q5 | Series |
91 | Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \left(\frac{3}{5}\right)^n \left(1-\frac{1}{n}\right) $ | \frac{3}{2}-\log \left(\frac{5}{2}\right) | 3/2 - log(5/2) | Original | ASyMOB
Series
Q6 | Series |
92 | Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \frac{(-1)^n}{2 n+5} $ | \frac{\pi }{4}-\frac{13}{15} | pi/4 - 13/15 | Original | ASyMOB
Series
Q7 | Series |
93 | Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty 2^{1-3 n} \cdot 3^{n+1} $ | \frac{18}{5} | 18/5 | Original | ASyMOB
Series
Q8 | Series |
94 | Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \frac{1}{n!} $ | e-1 | e - 1 | Original | ASyMOB
Series
Q9 | Series |
95 | Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \frac{1}{n^2-n-1} $ | -\frac{\left(\sqrt{5}-5\right) \pi \tan \left(\frac{\sqrt{5} \pi }{2}\right)}{5
\left(\sqrt{5}-1\right)} | sqrt(5)*pi*tan(sqrt(5)*pi/2)/5 | Original | ASyMOB
Series
Q10 | Series |
96 | Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \frac{n^3 + n^2 + n + 1}{n \cdot (n+1)^2 \cdot (n+2)^2} $ | \frac{5}{12} \left(\pi ^2-9\right) | 5*pi**2/12 - 15/4 | Original | ASyMOB
Series
Q11 | Series |
97 | Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \frac{n+1}{(n+2) \cdot (n+3) \cdot (n+4)} $ | \frac{5}{24} | 5/24 | Original | ASyMOB
Series
Q12 | Series |
98 | Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \frac{(-1)^n \cdot n+1}{(n+1) \cdot (n+2) \cdot (n+3)} $ | \frac{17}{6}-4 \log (2) | 17/6 - log(16) | Original | ASyMOB
Series
Q13 | Series |
99 | Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty (-2)^{-n} \left(n^2 + n + (-1)^n\right) $ | \frac{19}{27} | 19/27 | Original | ASyMOB
Series
Q14 | Series |
100 | Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \frac{(-1)^{n+1}+1}{n^2+1} $ | \frac{1}{2} \pi \tanh \left(\frac{\pi }{2}\right) | pi*tanh(pi/2)/2 | Original | ASyMOB
Series
Q15 | Series |
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