Index
stringlengths 1
5
| Challenge
stringlengths 41
1.55k
| Answer in Latex
stringclasses 122
values | Answer in Sympy
stringlengths 1
774
| Variation
stringclasses 31
values | Source
stringclasses 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 |
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