Index
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Challenge
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1.55k
Answer in Latex
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Answer in Sympy
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Variation
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31 values
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61 values
1
Compute the first 5 nonzero terms (not necessarily a quadratic polynomial) of the Maclaurin series of $ f(x) = 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
2
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left(n \cdot x^n\right)$ and $g(x) = \sum_{n=1}^\infty \left(n \cdot x^n\right)$
\sum_{n=2}^\infty\left(\frac{1}{6}\cdot n\cdot\left(n^2-1\right)\cdot x^n\right)
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
Original
U-Math sequences_series fb6418ae-3440-4258-9388-89d799fd859a
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
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
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
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
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
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
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
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
11
Integrate $ \int \sin(x)^4 \cdot \cos(x)^6 dx $
$C+\frac{1}{320}\cdot\left(\sin(2\cdot x)\right)^5+\frac{1}{128}\cdot\left(\frac{3\cdot x}{2}-\frac{\sin(4\cdot x)}{2}+\frac{\sin(8\cdot x)}{16}\right)$
C + 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
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
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
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
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 * ln(Abs((sqrt(x - 4) - sqrt(x + 4)) / (sqrt(x - 4) + sqrt(x + 4))))
Original
U-Math integral_calc 08c72d46-1abd-49e1-9c9c-ce509902be6e
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
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
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
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
20
Evaluate the integral: $ I = \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
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
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
23
Evaluate the limit. $ \lim_{x\to 1} \dfrac{x^2+11x-12}{\ln x}=$
13
13
Original
UGMathBench Calculus_-_single_variable_0508
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
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
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
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
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
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
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
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
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
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
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
35
Evaluate $\displaystyle{\int_0^4(2x-\sqrt{16-x^2})dx}$.
16 - 4 \pi
16 - 4*pi
Original
MathOdyssey Problem 325 from Calculus and Analysis - College Math
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
55
Solve the following integral: $\int_{0}^{\pi} \log(\sin(x)) dx$
-\pi \log (2)
-pi*log(2, E)
Original
OBMU 2019 - Q22
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
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
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
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
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
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
62
Compute up to degree 5 ($x^5$) the terms of the Maclaurin series of $ f(x) = A e^{B \sin(x)} $, where A and B are symbolic constants.
\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
63
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left(A n \cdot (F x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left(B n \cdot (F x)^n\right$
\sum_{n=2}^\infty \frac{1}{6} A B n \left(n^2-1\right) F^n \cdot x^n
A*B*Sum(F**n*x**n*n*(n**2 - 1), (n, 2, oo))/6
Symbolic-3
U-Math sequences_series fb6418ae-3440-4258-9388-89d799fd859a
64
Compute the first 6 nonzero terms of the Maclaurin series of $f(x) = F \left(\sin(A x) \cdot \cos\left(\frac{ B \pi }{ 4 }\right) + \cos(A x) \cdot \sin\left(\frac{ B \pi }{ 4 }\right)\right)$
\frac{1}{120} A^5 F x^5 \cos \left(\frac{\pi B}{4}\right)+\frac{1}{24} A^4 F x^4 \sin \left(\frac{\pi B}{4}\right)-\frac{1}{6} A^3 F x^3 \cos \left(\frac{\pi B}{4}\right)-\frac{1}{2} A^2 F x^2 \sin \left(\frac{\pi B}{4}\right)+A F x \cos \left(\frac{\pi B}{4}\right)+F \sin \left(\frac{\pi B}{4}\right)
F*(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
65
Compute the terms up to order 4 ($x^4$) of the Maclaurin series of $f(x) = F e^{A x} \cdot \cos(B x)$
F x^3 \left(\frac{A^3}{6}-\frac{A B^2}{2}\right)+\frac{1}{2} F x^2 \left(A^2-B^2\right)+F x^4 \left(\frac{A^4}{24}-\frac{A^2 B^2}{4}+\frac{B^4}{24}\right)+A F x+F
F*(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
66
Compute $\lim_{x \to 0}\frac{(2 \cos (F x)+4) \csc (F x)}{5 A (F x)^3}-\frac{6}{5 A (F x)^4}$
\frac{1}{A 150}
1/(150*A)
Symbolic-2
U-Math sequences_series 068e40ce-9108-4ef8-8ee5-0d1471ebbe43
67
Evaluate $ \lim_{x \to 0^+} A \left( \frac{ \tan\left( \frac{B x}{2} \right) }{ \frac{B x}{2} } \right)^{ \frac{F 3}{(B x)^2} } $
$A /cdot e^{\frac{F}{4}}$
A*e**(F/4)
Symbolic-3
U-Math differential_calc 363dd580-f1fc-4867-a6ef-db2a03139745
68
Evaluate $ \lim_{x \to A 5} \left( \frac{ 3 B x }{ x - 5 A }-\frac{ 3 B }{ \ln\left(\frac{ x }{ 5 A }\right) } \right)^{F} $
\left( \frac{3B}{2} \right)^F
(3*B/2)**F
Symbolic-3
U-Math differential_calc 2d799998-115a-489b-a48b-57090954303e
69
Evaluate $ \lim_{x \to \infty} \left(A x - A F x^2 \cdot \ln\left(1 + \frac{ 1 }{ F x }\right)\right)^(1 B) $
\left(\frac{A}{2 F}\right)^{B}
(A/(2*F))**B
Symbolic-3
U-Math differential_calc efdc4110-cf56-4f37-bf54-40fdd5d58145
70
Evaluate $ \lim_{x \to 0^+} F \left( \frac{\tan(A x)}{A x} \right)^{\frac{1 H}{3 B x^2}} $
F e^{\frac{A^{2} H}{9 B}}
F*e**((A**2*H)/((9*B)))
Symbolic-4
U-Math differential_calc 99a2304d-5d8e-4245-90da-a80651ca15d8
71
Evaluate $ \lim_{x \to 0} \left| F \left( \frac{-\sin(A x)}{A x} \right)^{\frac{1}{4 B x^2}} \right| $
\left( e^{-\frac{A^2}{6B}} \right)^{\frac{1}{4}} \left| F \right|
(exp(-A**2 / (6 * B)))**Rational(1, 4) * Abs(F)
Symbolic-3
U-Math differential_calc 84c6a419-c103-41d5-aad5-dd8e690c6e88
72
Integrate $ \int B \sin(F x)^4 \cdot \cos(F x)^6 dx $
\frac{\frac{B \sin{\left(2 F x \right)}}{512} - \frac{B \sin{\left(4 F x \right)}}{256} - \frac{B \sin{\left(6 F x \right)}}{1024} + \frac{B \sin{\left(8 F x \right)}}{2048} + \frac{B \sin{\left(10 F x \right)}}{5120} + \frac{F x \left(256 A + 3 B\right)}{256}}{F}
(3*B*x)/256 + (B*sin(2*F*x))/(512*F) - (B*sin(4*F*x))/(256*F) - (B*sin(6*F*x))/(1024*F) + (B*sin(8*F*x))/(2048*F) + (B*sin(10*F*x))/(5120*F)
Symbolic-2
U-Math integral_calc 0c0ba3db-1470-4c36-975c-91ff5f51986f
73
Solve the following integral. Assume A,B,F,G are real and positive. $ \int \frac{A \sqrt[5]{x} + B x^{4/5} + F x^{6/5}}{x \left(1 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-F G)+2 G (F G-B) \log \left(G+x^{2/5}\right)+F x^{4/5}\right)
(5/4)*(F*x**(4/5) + ((2*(x**(2/5)*(B - F*G)) + (4*(A*atan(x**(1/5)/(sqrt(G)))))/(sqrt(G))) + 2*(G*(-B + F*G)*log(G + x**(2/5), E))))
Symbolic-4
U-Math integral_calc 126c4165-b3d5-4470-8412-08e79d9821cf
74
Solve the following integral. Assume A,B,F are real and positive $ \int \frac{A \csc ^7(F x) \sec (F x)}{1 B} dx $
-\frac{A \left(2 \csc ^6(F x)+3 \csc ^4(F x)+6 \csc ^2(F x)+12 (\log (\cos (F x))-\log (\sin (F x)))\right)}{12 B F}
-A*(-12*log(sin(F*x)) + 12*log(cos(F*x)) + 2*csc(F*x)**6 + 3*csc(F*x)**4 + 6*csc(F*x)**2)/(12*B*F)
Symbolic-3
U-Math integral_calc 00f6affb-905a-4109-a78e-2dde7a0b83accf
75
Solve the following integral. Assume A,B,F are real and positive. $ \int -\frac{2 A \sqrt{4 B + (F x)^2}}{ (F x)^4} dx $
\frac{A \left(4 B+F^2 x^2\right)^{3/2}}{6 B F^4 x^3}
A*(4*B + F**2*x**2)**(3/2)/(6*B*F**4*x**3)
Symbolic-3
U-Math integral_calc 05ea9929-8cbb-432b-bbbb-ec1e74c9f401
76
Solve the following integral. Assume A,B,F are real and positive. $ \int \left(\frac{B (4 A + F x)}{F x - 4 A}\right)^{3/2} dx $
\frac{B \sqrt{\frac{B (4 A+F x)}{F x-4 A}} \left(\sqrt{4 A+F x} (F x-20 A)+24 A \sqrt{F x-4 A} \tanh ^{-1}\left(\frac{\sqrt{4 A+F x}}{\sqrt{F x-4 A}}\right)\right)}{F \sqrt{4 A+F x}}
B*sqrt(-B*(4*A + F*x)/(4*A - F*x))*(24*A*sqrt(-4*A + F*x)*atanh(sqrt(4*A + F*x)/sqrt(-4*A + F*x)) + (-20*A + F*x)*sqrt(4*A + F*x))/(F*sqrt(4*A + F*x))
Symbolic-3
U-Math integral_calc 08c72d46-1abd-49e1-9c9c-ce509902be6e
77
Solve the following integral. Assume A,B,F,G are real and positive. $ \int \frac{ -1 A }{B (F x)^2 \cdot \left(3 G + (F x)^3\right)^{\frac{ 5 }{ 3 }} } dx $
\frac{A \left(F^3 x^3+2 G\right)}{6 B F^2 G^2 x \left(F^3 x^3+3 G\right)^{2/3}}
A*(F**3*x**3 + 2*G)/(6*B*F**2*G**2*x*(F**3*x**3 + 3*G)**(2/3))
Symbolic-4
U-Math integral_calc 4c1292e1-d4b3-4acf-afaf-eaac62f2662d
78
Solve the following integral. Assume A,B,F,G,H are real and positive. $ \int \frac{\sqrt{4 A x-5 B}+4 F x}{5 G \sqrt[4]{4 A x - 5 B} + H (4 A x - 5 B)^{3/4}} dx $
\frac{\frac{\sqrt{H} \left(20 A^2 H^2 x+375 F G^2 \sqrt{4 A x-5 B}+5 B H \left(12 F H \sqrt{4 A x-5 B}-5 A H+25 F G\right)+A H \left(12 F H x \sqrt{4 A x-5 B}-75 G \sqrt{4 A x-5 B}-100 F G x\right)\right)}{\sqrt[4]{4 A x-5 B}}-75 \sqrt{5} \sqrt{G} \left(-A G H+B F H^2+5 F G^2\right) \tan ^{-1}\left(\frac{\sqrt{H} \sqrt[4]{4 A x-5 B}}{\sqrt{5} \sqrt{G}}\right)}{15 A^2 H^{7/2}}
(75*sqrt(5)*sqrt(G)*(4*A*x - 5*B)**(1/4)*(A*G*H - B*F*H**2 - 5*F*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*F*G*x + 12*F*H*x*sqrt(4*A*x - 5*B) - 75*G*sqrt(4*A*x - 5*B)) + 5*B*H*(-5*A*H + 25*F*G + 12*F*H*sqrt(4*A*x - 5*B)) + 375*F*G**2*sqrt(4*A*x - 5*B)))/(15*A**2*H**(7/2)*(4*A*x - 5*B)**(1/4))
Symbolic-5
U-Math integral_calc 147944c5-b782-48c5-a664-d66deb92d9a7
79
Solve the following integral. Assume A,B,F are real and positive. $ \int \frac{3 A \csc ^7(2 F x) \sec (2 F x)}{1 B} dx $
-\frac{A \left(2 \csc ^6(2 F x)+3 \csc ^4(2 F x)+6 \csc ^2(2 F x)+12 (\log (\cos (2 F x))-\log (\sin (2 F x)))\right)}{8 B F}
-A*(-12*log(sin(2*F*x)) + 12*log(cos(2*F*x)) + 2*csc(2*F*x)**6 + 3*csc(2*F*x)**4 + 6*csc(2*F*x)**2)/(8*B*F)
Symbolic-3
U-Math integral_calc 1db212f0-2fac-410d-969d-fe3b5b55d076
80
Solve the following integral. Assume A,F are real and positive. $ \int A \csc^5 (8 F x) dx $
-\frac{A \left(\csc ^4(4 F x)+6 \csc ^2(4 F x)-\sec ^4(4 F x)-6 \sec ^2(4 F x)+24 (\log (\cos (4 F x))-\log (\sin (4 F x)))\right)}{512 F}
-A*(-24*log(sin(4*F*x)) + 24*log(cos(4*F*x)) + csc(4*F*x)**4 + 6*csc(4*F*x)**2 - sec(4*F*x)**4 - 6*sec(4*F*x)**2)/(512*F)
Symbolic-2
U-Math integral_calc 275f7ceb-f331-4a3f-96ec-346e6d81b32a
81
Solve the following integral. Assume A,B,F are real and positive. $ \int \cos (2 F x) \left(A (F x)^3+3 B \right) dx $
\frac{2 \sin (2 F x) \left(A F x \left(2 F^2 x^2-3\right)+6 B\right)+3 A \left(2 F^2 x^2-1\right) \cos (2 F x)}{8 F}
((2*A*F*x*(2*F**2*x**2 - 3) + 12*B)*sin(2*F*x) + 3*A*(2*F**2*x**2 - 1)*cos(2*F*x))/(8*F)
Symbolic-3
U-Math integral_calc 47a11349-0386-4969-9263-d3cdfcc98cb9
82
Use factoring to calculate the following limit. Assume A,B,F are real and positive. $ \lim_{x \rightarrow K} \frac{(F x)^{4 B} - K^{4 B}}{A \left((F x)^{5 B}- K^{5 B}\right)} $
\frac{4 K^{-B}}{5 A}
4/(5*A*K**B)
Symbolic-3
UGMathBench Calculus_-_single_variable_0016
83
Calculate the following limit. Assume A,B,F are real and positive. $ \frac{1 B - B \cos (10 F x)}{A \cos ^2(6 F x) - 1 A} $
-\frac{25 B}{18 A}
-25*B/(18*A)
Symbolic-3
UGMathBench Calculus_-_single_variable_0022
84
Calculate the following limit. Assume A,B,F are real and positive. $ \frac{A (F x)^2+11 A F x - 12 A}{B \log (F x)} $
\frac{13 A}{B}
13*A/B
Symbolic-3
UGMathBench Calculus_-_single_variable_0508
85
Calculate the following limit. Assume A,F are real and A>1. $ \lim\limits_{x\to+\infty} 4^{-\frac{1}{F x}} \left(\frac{(4 A)^{F x} + (6 A)^{F x} }{1 A}\right)^{\frac{1}{F x}} $
6 A
6*A
Symbolic-2
UGMathBench Calculus_-_single_variable_0512
86
Calculate the following integral. Assume A,B, F are real and positive. $\int_{2 B}^{\infty} 3 A (F x)^2 e^{- (F x)^3} dx=$
\frac{A e^{-8 B^3 F^3}}{F}
A/(F*e**(8*B**3*F**3))
Symbolic-3
UGMathBench Calculus_-_single_variable_0592
87
Evaluate the indefinite integral. Assume A, F are real and positive. $\int A \tan ^3(F x) \sec ^9(F x) dx$
\frac{A \sec ^9(F x) \left(9 \sec ^2(F x)-11\right)}{99 F}
A*(9*sec(F*x)**2 - 11)*sec(F*x)**9/(99*F)
Symbolic-2
UGMathBench Calculus_-_single_variable_0604
88
Evaluate the indefinite integral. Assume A, F are real and positive. $\int 208 A \cos ^4(16 F x) dx$
\frac{13 A (192 F x+8 \sin (32 F x)+\sin (64 F x))}{32 F}
13*A*(192*F*x + 8*sin(32*F*x) + sin(64*F*x))/(32*F)
Symbolic-2
UGMathBench Calculus_-_single_variable_0606
89
Evaluate the integral. Assume A, B, F, G are real and positive. $ \int \frac{-38 A+10 B (F x)^2-48 F G x}{(F x)^3-5 (F x)^2-8 F x+48} dx $
\frac{2 \left(\frac{7 (19 A-80 B+96 G)}{F x-4}+(19 A+200 B-72 G) \log (4-F x)+(-19 A+45 B+72 G) \log (F x+3)\right)}{49 F}
2*(133*A - 560*B + 672*G + (F*x - 4)*((-19*A + 45*B + 72*G)*log(F*x + 3) + (19*A + 200*B - 72*G)*log(-F*x + 4)))/(49*F*(F*x - 4))
Symbolic-4
UGMathBench Calculus_-_single_variable_0612
90
Evaluate the integral. Assume A,B,F are real and positive. $ \int A e^{F x} \sqrt{64 B-e^{2 F x}} \;dx$
\frac{A \left(e^{F x} \sqrt{64 B-e^{2 F x}}+64 B \tan ^{-1}\left(\frac{e^{F x}}{\sqrt{64 B-e^{2 F x}}}\right)\right)}{2 F}
A*(64*B*atan(e**(F*x)/sqrt(64*B - e**(2*F*x))) + e**(F*x)*sqrt(64*B - e**(2*F*x)))/(2*F)
Symbolic-3
UGMathBench Calculus_-_single_variable_0624
91
Evaluate the following limit. Assume A,B,F are real and positive. $\lim_{x \to 0} \frac{-\frac{9}{2} (1 B)^2 (F x)^6+3 B F^3 x^3+e^{-3 B (F x)^3}-1}{12 A (F x)^9} $
-\frac{3 B^3}{8 A}
-3*B**3/(8*A)
Symbolic-3
UGMathBench Calculus_-_single_variable_0939
92
Solve the following first-order differential equation: Assume A,B,F,G are real and positive. $ A y'(x)+2 B y(x)=F e^{-x}, \quad y(0)=1 G.$
\frac {e^{-\frac{2 B x}{A}} \left(F \left(-e^{x \left(\frac{2 B}{A}-1\right)}\right)+A G-2 B G+F\right)}{A-2 B}
(A*G - 2*B*G + F + F*(-e**x*((-A + 2*B)/A)))/(e**(2*B*x/A)*(A - 2*B))
Symbolic-4
MathOdyssey Problem 340 from Differential Equations - College Math
93
Consider the differential equation $A y'(x)=B x y(x)$. Find the value of $y(\sqrt{2})$ given that $y(0) = 2 F$. Assume A,B,F are real and positive.
2 F e^{\frac{B}{A}}
2*F*e**(B/A)
Symbolic-3
MathOdyssey Problem 339 from Differential Equations - College Math
94
Evaluate the following limit: $ \lim_{x \to \infty} \sqrt{-(1 B) + H (F x)^2 + 2 F G x}-\sqrt{3 A + H (F x)^2} .$ Assume A,B,F,G,H are real and positive.
\frac{G}{\sqrt{H}}
G/sqrt(H)
Symbolic-5
MathOdyssey Problem 315 from Calculus and Analysis - College Math
95
Evaluate $\lim\limits_{x\to \frac{4 B}{1 F}} \frac{A (F x - 4 B)}{\sqrt{F x}-2 \sqrt{1 B}} $. Assume A,B,F are real and positive.
4 A \sqrt{B}
4*A*sqrt(B)
Symbolic-3
MathOdyssey Problem 317 from Calculus and Analysis - College Math
96
Evaluate $\int_0^(4 B) (2 A x - \sqrt{(4 B F)^2 - (F x)^2}) dx$. Assume A,B,F are real and positive.
\frac{1}{4} B^2 (4 A-\pi F)
B**2*(A - F*pi/4)
Symbolic-3
MathOdyssey Problem 325 from Calculus and Analysis - College Math
97
Evaluate the series $\sum\limits_{x=1}^\infty \frac{1 A}{B (1 F + x) (1 F+x+2)} $. Assume A,B,F are real and positive.
\frac{A (2 F+3)}{2 B (F+1) (F+2)}
A*(2*F + 3)/(2*(F + 2)*B*(F + 1))
Symbolic-3
MathOdyssey Problem 326 from Calculus and Analysis - College Math
98
Evaluate the limit $\lim\limits_{x \to 0} \frac{(A x+1)^{\frac{1}{A x}}-e}{B 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
99
Evaluate the series $\sum\limits_{n=0}^\infty \frac{ \left(\frac{1}{2 B}\right)^{A (2 n+1)}}{F (2 n+1)} $. Assume A,B,F are real and positive.
\frac{\tanh ^{-1}\left(2^{-A} \left(\frac{1}{B}\right)^A\right)}{F}
atanh((1/(2*B))**A)/F
Symbolic-3
MathOdyssey Problem 328 from Calculus and Analysis - College Math
100
Evaluate the limit $\lim\limits_{n\to\infty}\sum\limits_{k=0}^{n-1}\frac{1 A}{B \sqrt{F n^2-k^2}}$ Assume A,B, $F \ge 1$ are real and positive.
\frac{A}{B} \arcsin\left(\frac{1}{\sqrt{F}}\right)
A*asin(1/sqrt(F))/B
Symbolic-3
MathOdyssey Problem 329 from Calculus and Analysis - College Math