toloka/u-math huggingface dataset:
Subject: multivariable_calculus

Problem: Compute the second order derivative $\frac{d ^2y}{ d x^2}$ for the parametrically defined function
$$
x = \frac{ 1 }{ 2 } \cdot \left(\cos(t)\right)^3, \quad y = 6 \cdot \left(\sin(t)\right)^3
$$

Answer: $\frac{d ^2y}{ d x^2}$ = $\frac{8}{\left(\cos(t)\right)^4\cdot\sin(t)}$
Subject: precalculus_review

Problem: Evaluate the definite integral. Express answer in exact form whenever possible:
$$
\int_{0}^\pi \left(\cos(99 \cdot x) \cdot \sin(101 \cdot x)\right) \, dx
$$

Answer: $\int_{0}^\pi \left(\cos(99 \cdot x) \cdot \sin(101 \cdot x)\right) \, dx$ = $0$
Subject: precalculus_review

Problem: Calculate the following indefinite integral:
$$
\int x \cdot \left(a^x\right)^x \, dx
$$
for $a > 0$ and $a \ne 1$.

Answer: $\int x \cdot \left(a^x\right)^x \, dx$ = $\frac{a^{x^2}}{2\cdot\ln(a)}+C$
Subject: differential_calc

Problem: Find the local extrema of the function $f(x) = 2 \cdot \left(x^2\right)^{\frac{ 1 }{ 3 }} - x^2$ using the First Derivative Test.

Answer: Local maxima: $x=-\sqrt[4]{\frac{8}{27}}$, $x=\sqrt[4]{\frac{8}{27}}$

Local minima: $x=0$
Subject: algebra

Problem: Solve the equation for $x$:

$12 - 5 \cdot (x + 3) = 2 \cdot x - 5$

Answer: The final answer: $x=\frac{2}{7}$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ x-\sqrt[3]{x^2}-\sqrt[6]{x} }{ x \cdot \left(4+\sqrt[3]{x}\right) } \, dx
$$

Answer: $\int \frac{ x-\sqrt[3]{x^2}-\sqrt[6]{x} }{ x \cdot \left(4+\sqrt[3]{x}\right) } \, dx$ = $C+\frac{3}{2}\cdot\sqrt[3]{x^2}+60\cdot\ln\left(\left|4+\sqrt[3]{x}\right|\right)-3\cdot\arctan\left(\frac{1}{2}\cdot\sqrt[6]{x}\right)-15\cdot\sqrt[3]{x}$
Subject: algebra

Problem: Using the Rational Zero Theorem, list all possible rational zeros of the following polynomial:

$p(x) = 2 \cdot x^3 + x - 1$

Answer: Possible rational zeros are $1$, $-1$, $\frac{1}{2}$, $-\frac{1}{2}$
Subject: sequences_series

Problem: Find the radius of convergence of the series:
$$
\sum_{n=1}^\infty \left(\left(\frac{ n-1 }{ 2 \cdot n+3 }\right)^n \cdot x^n\right)
$$

Answer: $R$ = $2$
Subject: precalculus_review

Problem: Solve the following system of equations:

1. $\sin(x) \cdot \sin(y) = \frac{ \sqrt{3} }{ 4 }$
2. $\cos(x) \cdot \cos(y) = \frac{ \sqrt{3} }{ 4 }$

Answer: The final answer: $x=\frac{\pi}{3}+\frac{\pi}{2}\cdot(2\cdot n+k)$, $x=\frac{\pi}{6}+\frac{\pi}{2}\cdot(2\cdot n+k)$, $y=\frac{\pi}{6}+\frac{\pi}{2}\cdot(k-2\cdot n)$, $y=\frac{\pi}{3}+\frac{\pi}{2}\cdot(k-2\cdot n)$
Subject: precalculus_review

Problem: Solve $x^2 - 4 \cdot x - 6 = \sqrt{2 \cdot x^2 - 8 \cdot x + 12}$.

Answer: The final answer: $x=-2 \lor x=6$
Subject: precalculus_review

Problem: Calculate the following integral:
$$
\int \log_{a^x}(b) \, dx
$$
for $a > 0$, $b > 0$, $a \ne 1$, $b \ne 1$, $x \ne 0$.

Answer: $\int \log_{a^x}(b) \, dx$ = $\frac{\ln(x)\cdot\ln(b)}{\ln(a)}+C$
Subject: sequences_series

Problem: Suppose that we have a series: $1-\frac{ 1 }{ 2 }+\frac{ 1 }{ 3 }-\cdots+(-1)^{n+1} \cdot \frac{ 1 }{ n }+\cdots$ and its sum is equal to $\ln(2)$. Find the sum of the series: $1-\frac{ 1 }{ 2 }-\frac{ 1 }{ 4 }+\frac{ 1 }{ 3 }-\frac{ 1 }{ 6 }-\frac{ 1 }{ 8 }+\frac{ 1 }{ 5 }-\frac{ 1 }{ 10 }-\frac{ 1 }{ 12 }+\cdots$.

Answer: The final answer: $S=\frac{1}{2}\cdot\ln(2)$
Subject: integral_calc

Problem: Compute the integral:
$$
3 \cdot \int \left(\cos\left(\frac{ x }{ 6 }\right)\right)^6 \, dx
$$

Answer: $3 \cdot \int{\left(\cos\left(\frac{ x }{ 6 }\right)\right)^6 d x}$ = $C+\frac{9}{2}\cdot\sin\left(\frac{x}{3}\right)+\frac{15}{16}\cdot x+\frac{27}{32}\cdot\sin\left(\frac{2\cdot x}{3}\right)-\frac{3}{8}\cdot\left(\sin\left(\frac{x}{3}\right)\right)^3$
Subject: differential_calc

Problem: Evaluate $\lim_{x \to 0^{+}} \left( \left( \frac{ \tan(4 \cdot x) }{ 4 \cdot x } \right)^{\frac{ 1 }{ 2 \cdot x^2 }} \right)$ using L'Hopital's Rule.

Answer: $\lim_{x \to 0^{+}} \left( \left( \frac{ \tan(4 \cdot x) }{ 4 \cdot x } \right)^{\frac{ 1 }{ 2 \cdot x^2 }} \right)$ = $e^{\frac{8}{3}}$
Subject: precalculus_review

Problem: Given $k$ as an integer and $a \ne \pi \cdot k$, evaluate $P=\cos(a) \cdot \cos(2 \cdot a) \cdot \cos(4 \cdot a) \ldots \cos\left(2^n \cdot a\right)$.

Answer: The final answer: $P=\frac{1}{2^{n+1}}\cdot\frac{\sin\left(2^{n+1}\cdot a\right)}{\sin(a)}$
Subject: algebra

Problem: Multiply the rational expressions and express the product in simplest form:

$$
\frac{ 2 \cdot d^2+9 \cdot d-35 }{ d^2+10 \cdot d+21 } \cdot \frac{ 3 \cdot d^2+2 \cdot d-21 }{ 3 \cdot d^2+14 \cdot d-49 }
$$

Answer: The final answer: $\frac{2\cdot d-5}{d+7}$
Subject: integral_calc

Problem: Solve the integral:
$$
\int \left(\frac{ x+6 }{ x-6 } \right)^{\frac{ 3 }{ 2 }} \, dx
$$

Answer: $\int \left(\frac{ x+6 }{ x-6 } \right)^{\frac{ 3 }{ 2 }} \, dx$ = $C+\sqrt{\frac{x+6}{x-6}}\cdot(x-30)-18\cdot\ln\left(\left|\frac{\sqrt{x-6}-\sqrt{x+6}}{\sqrt{x-6}+\sqrt{x+6}}\right|\right)$
Subject: sequences_series

Problem: Find the radius of convergence of the series:
$$
\sum_{n=1}^\infty \left( \left(n^{\frac{ 1 }{ n }}-1\right)^n \cdot x^n \right)
$$

Answer: $R$ = $\infty$
Subject: precalculus_review

Problem: Solve the differential equation: $\frac{ d y }{d x} = \sin(x)^2$. The curve passes through the point $(0,0)$.

Answer: $f(x)$ = $\frac{x}{2}-\frac{\sin(2\cdot x)}{4}$
Subject: multivariable_calculus

Problem: A small appliances company makes toaster ovens and pizza cookers, and have noticed their customer base's purchasing habits give them the price-demand equations given below, where $p$ is the price and $x$ is the quantity of toaster ovens, $q$ is the price and $y$ is the quantity of pizza cookers. The Cost function is $C(x,y) = 160 + 12 \cdot x + 24 \cdot y$.

$p = 50 - 4 \cdot x + y$ and $q = 100 + x - 5 \cdot y$

1. Construct the 2-product Revenue function, $R(x,y)$
2. Find $R(8,10)$
3. Construct the 2-product Profit function, $P(x,y)$
4. Find $P(8,10)$

Answer: 1. $R(x,y)$ = $R(x,y)=50\cdot x+100\cdot y+2\cdot x\cdot y-4\cdot x^2-5\cdot y^2$
2. $R(8,10)$ = $804$
3. $P(x,y)$ = $P(x,y)=38\cdot x+76\cdot y+2\cdot x\cdot y-160-4\cdot x^2-5\cdot y^2$
4. $P(8,10)$ = $308$
Subject: precalculus_review

Problem: 1. Find the inverse function of $f(x) = \sqrt{x-1}$
2. Find the domain of the inverse function
3. Find the range of the inverse function

Answer: 1. The inverse function is: $x^2+1$
2. The domain of the inverse function is: $x\ge0$
3. The range of the inverse function is: $y\ge1$
Subject: differential_calc

Problem: Find the derivative of the function $y = \frac{ 3 \cdot \csc(x) - 4 \cdot \sin(x) }{ 8 \cdot \left(\cos(x)\right)^5 } - \frac{ 76 }{ 5 } \cdot \cot(3 \cdot x)$.

Answer: $y'$ = $\frac{228}{5\cdot\left(\sin(3\cdot x)\right)^2}+\frac{16\cdot\left(\cos(x)\right)^6-5\cdot\left(\cos(x)\right)^4-3\cdot\left(\cos(x)\right)^6\cdot\left(\csc(x)\right)^2}{8\cdot\left(\cos(x)\right)^{10}}$
Subject: sequences_series

Problem: Find the Fourier series of the function $f(x) = \frac{ 1 }{ 4 } \cdot x$ in the interval $[-3,3]$.

Answer: The Fourier series is: $\sum_{n=1}^\infty\left(\frac{\frac{3}{2}\cdot(-1)^{n+1}}{\pi\cdot n}\cdot\sin\left(\frac{\pi\cdot n\cdot x}{3}\right)\right)$
Subject: integral_calc

Problem: Find the area bounded by the curves $y = 3 \cdot x + 3$, $y = 3 \cdot \cos(x)$, and $y = 0$.

Answer: Area: $\frac{9}{2}$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ 1 }{ \left(\cos(5 \cdot x)\right)^3 } \, dx
$$

Answer: $\int \frac{ 1 }{ \left(\cos(5 \cdot x)\right)^3 } \, dx$ = $C+\frac{\sin(5\cdot x)}{10\cdot\left(\cos(5\cdot x)\right)^2}+\frac{1}{10}\cdot\ln\left(\left|\tan\left(\left(\frac{5}{2}\right)\cdot x+\frac{\pi}{4}\right)\right|\right)$
Subject: sequences_series

Problem: Find the domain of convergence of the series $\sum_{n=1}^\infty \left(\frac{ x^n }{ n \cdot 10^{n-1} }\right)$.

Answer: The final answer: $-10\le x<10$
Subject: integral_calc

Problem: Compute the integral:
$$
2 \cdot \int \left(\cos\left(\frac{ x }{ 4 }\right)\right)^6 \, dx
$$

Answer: $2 \cdot \int{\left(\cos\left(\frac{ x }{ 4 }\right)\right)^6 d x}$ = $C+2\cdot\sin\left(\frac{x}{2}\right)+\frac{3}{8}\cdot\sin(x)+\frac{5}{8}\cdot x-\frac{1}{6}\cdot\left(\sin\left(\frac{x}{2}\right)\right)^3$
Subject: algebra

Problem: Use synthetic division to find the quotient and remainder. Ensure the equation is in the form required by synthetic division:

$$
\frac{ 4 \cdot x^3-12 \cdot x^2-5 \cdot x-1 }{ 2 \cdot x+1 }
$$

Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor. Solve on a paper, if it is more convenient for you.

Answer: Quotient: $2\cdot x^2-7\cdot x+1$ Remainder: $-2$
Subject: precalculus_review

Problem: Find the inverse of the function $f(x) = \frac{ 4-2 \cdot \sqrt{x} }{ 4+2 \cdot \sqrt{x} }$.

Answer: The final answer: $\frac{4\cdot(1-x)^2}{(x+1)^2}$
Subject: sequences_series

Problem: Using the series expansion for the function $(1+x)^m$, calculate approximately $\sqrt[4]{15}$ with an accuracy of 0.0001.

Answer: The final answer: $1.968$
Subject: differential_calc

Problem: What are the points of inflection of the graph of $f(x) = \frac{ x+1 }{ x^2+1 }$?

Answer: The final answer: $x_1=1 \land x_2=-2+\sqrt{3} \land x_3=-2-\sqrt{3}$
Subject: differential_calc

Problem: Find the local minimum and local maximum values of the function $f(x) = \frac{ x^4 }{ 2 } - \frac{ 20 }{ 3 } \cdot x^3 + 24 \cdot x^2 + 13$.

Answer: The point(s) where the function has a local minimum: $P(0,13)$, $P(6,85)$
The point(s) where the function has a local maximum: $P\left(4,\frac{295}{3}\right)$
Subject: differential_calc

Problem: Evaluate $\lim_{x \to 0}\left(\left(\frac{ \sin(4 \cdot x) }{ 4 \cdot x }\right)^{\frac{ 1 }{ 3 \cdot x^2 }}\right)$.

Answer: $\lim_{x \to 0}\left(\left(\frac{ \sin(4 \cdot x) }{ 4 \cdot x }\right)^{\frac{ 1 }{ 3 \cdot x^2 }}\right)$ = $e^{-\frac{8}{9}}$
Subject: multivariable_calculus

Problem: Find partial derivatives of the function $r = \sqrt{a \cdot x^2 - b \cdot y^2}$.

Answer: $\frac{\partial}{\partial x}(r)$ = $\frac{a\cdot x}{\sqrt{a\cdot x^2-b\cdot y^2}}$
$\frac{\partial}{\partial a}(r)$ = $\frac{x^2}{2\cdot\sqrt{a\cdot x^2-b\cdot y^2}}$
$\frac{\partial}{\partial y}(r)$ = $-\frac{b\cdot y}{\sqrt{a\cdot x^2-b\cdot y^2}}$
$\frac{\partial}{\partial b}(r)$ = $-\frac{y^2}{2\cdot\sqrt{a\cdot x^2-b\cdot y^2}}$
Subject: sequences_series

Problem: Find the Fourier series of the function $f(x) = \frac{ -1 }{ 3 } \cdot x$ in the interval $[-3,3]$.

Answer: The Fourier series is: $\sum_{n=1}^\infty\left(\frac{2\cdot(-1)^n}{\pi\cdot n}\cdot\sin\left(\frac{\pi\cdot n\cdot x}{3}\right)\right)$
Subject: algebra

Problem: Solve the following equations:

1. $2 m - 8 = -28$
2. $\frac{ x }{ 9 } - 3 = 8$
3. $12 m + 20 = -40$
4. $-\frac{ x }{ 3 } + 5 = 21$
5. $8 r - 27 = -19$
6. $6 + \frac{ k }{ 3 } = 33$
7. $15 = -4 y - 9$
8. $8 w + 4 = -36$

Answer: The solutions to the given equations are:
1. $m=-10$
2. $x=99$
3. $m=-5$
4. $x=-48$
5. $r=1$
6. $k=81$
7. $y=-6$
8. $w=-5$
Subject: precalculus_review

Problem: Solve the following inequality by using rules of the absolute value function. Express your answer as an interval or union of intervals.

$\left| |x+3| + |x-2| \right| \le 9$

Answer: The final answer: $[-5,4]$
Subject: sequences_series

Problem: Find the Fourier series of the function $u = \left| \frac{ \sin\left( \frac{ x }{ 2 } \right) }{ 2 } \right|$ in the interval $[-\pi, \pi]$.

Answer: The Fourier series is: $\frac{1}{\pi}-\frac{2}{\pi}\cdot\sum_{n=1}^\infty\left(\frac{1}{\left(4\cdot n^2-1\right)}\cdot\cos(n\cdot x)\right)$
Subject: differential_calc

Problem: Find the derivative of the 25th order $y^{(25)}$ for a function $y = 2 \cdot x^2 \cdot \sin(x)$.

Answer: $y^{(25)}$ = $\left(2\cdot x^2-1200\right)\cdot\cos(x)+100\cdot x\cdot\sin(x)$
Subject: differential_calc

Problem: Find the second derivative $\frac{d ^2y}{ d x^2}$ of the function $x = 5 \cdot \left(\cos(t)\right)^3$, $y = 6 \cdot \left(\sin(3 \cdot t)\right)^3$.

Answer: $\frac{d ^2y}{ d x^2}$ = $\frac{-15\cdot\left(\cos(t)\right)^2\cdot\sin(t)\cdot\left(324\cdot\sin(3\cdot t)-486\cdot\left(\sin(3\cdot t)\right)^3\right)-\left(30\cdot\cos(t)-45\cdot\left(\cos(t)\right)^3\right)\cdot54\cdot\left(\sin(3\cdot t)\right)^2\cdot\cos(3\cdot t)}{\left(-15\cdot\left(\cos(t)\right)^2\cdot\sin(t)\right)^3}$
Subject: algebra

Problem: Rewrite the quadratic expression $x^2 - 10 \cdot x + 7$ by completing the square.

Answer: $x^2 - 10 \cdot x + 7$ = $(x-5)^2-18$
Subject: precalculus_review

Problem: Solve $\sin(x) + 7 \cdot \cos(x) + 7 = 0$.

Answer: The final answer: $x=2\cdot\pi\cdot k-2\cdot\arctan(7) \lor x=\pi+2\cdot\pi\cdot k$
Subject: differential_calc

Problem: Sketch the curve:  
$$
y = \frac{ x^3 }{ 5 \cdot (x+2)^2 }
$$
Provide the following:

1. The domain (in interval notation)
2. Vertical asymptotes
3. Horizontal asymptotes
4. Slant asymptotes
5. Intervals where the function is increasing
6. Intervals where the function is decreasing
7. Intervals where the function is concave up
8. Intervals where the function is concave down
9. Points of inflection

Answer: 1. The domain (in interval notation): $(-1\cdot\infty,-2)\cup(-2,\infty)$
2. Vertical asymptotes: $x=-2$
3. Horizontal asymptotes: None
4. Slant asymptotes: $y=\frac{x}{5}-\frac{4}{5}$
5. Intervals where the function is increasing: $(-\infty,-6)$, $(-2,0)$, $(0,\infty)$
6. Intervals where the function is decreasing: $(-6,-2)$
7. Intervals where the function is concave up: $(-\infty,-6)$, $(-2,0)$, $(0,\infty)$
8. Intervals where the function is concave down: $(-\infty,-2)$, $(-2,0)$
9. Points of inflection: $P(0,0)$
Subject: differential_calc

Problem: Sketch the curve:  

$y = 5 \cdot x^2 - 2 \cdot x^4 - 3$  

Submit as your final answer:

1. The domain (in interval notation)
2. Vertical asymptotes (Leave blank if there are no vertical asymptotes)
3. Horizontal asymptotes (Leave blank if there are no horizontal asymptotes)
4. Slant asymptotes (Leave blank if there are no slant asymptotes)
5. Intervals where the function is increasing (Leave blank if there are no such intervals)
6. Intervals where the function is decreasing (Leave blank if there are no such intervals)
7. Intervals where the function is concave up (Leave blank if there are no such intervals)
8. Intervals where the function is concave down (Leave blank if there are no such intervals)
9. Points of inflection (Leave blank if there are no points of inflection)

Answer: 1. The domain (in interval notation): $(-1\cdot\infty,\infty)$
2. Vertical asymptotes: None
3. Horizontal asymptotes: None
4. Slant asymptotes: None
5. Intervals where the function is increasing: $\left(0,\frac{\sqrt{5}}{2}\right)$, $\left(-\infty,-\frac{\sqrt{5}}{2}\right)$
6. Intervals where the function is decreasing: $\left(-\frac{\sqrt{5}}{2},0\right)$, $\left(\frac{\sqrt{5}}{2},\infty\right)$
7. Intervals where the function is concave up: $\left(-\frac{\sqrt{5}}{2\cdot\sqrt{3}},\frac{\sqrt{5}}{2\cdot\sqrt{3}}\right)$
8. Intervals where the function is concave down: $\left(-\infty,-\frac{\sqrt{5}}{2\cdot\sqrt{3}}\right)$, $\left(\frac{\sqrt{5}}{2\cdot\sqrt{3}},\infty\right)$
9. Points of inflection: $P\left(\frac{\sqrt{5}}{2\cdot\sqrt{3}},-\frac{91}{72}\right)$, $P\left(-\frac{\sqrt{5}}{2\cdot\sqrt{3}},-\frac{91}{72}\right)$
Subject: integral_calc

Problem: The velocity of a bullet from a rifle can be approximated by $v(t) = 6400 \cdot t^2 - 6505 \cdot t + 2686$ where $t$ is seconds after the shot and $v$ is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: $0 \le t \le 0.5$. What is the total distance the bullet travels in $0.5$ sec?

Answer: The total distance is: $796.54166667$
Subject: integral_calc

Problem: Solve the integral:
$$
\int -16 \cdot \sin(-3 \cdot x)^4 \cdot \cos(-3 \cdot x)^2 \, dx
$$

Answer: $\int -16 \cdot \sin(-3 \cdot x)^4 \cdot \cos(-3 \cdot x)^2 \, dx$ = $-x+\frac{\sin(12\cdot x)}{12}+\frac{\sin(6\cdot x)}{12}-\frac{\sin(18\cdot x)}{36}+C$
Subject: precalculus_review

Problem: Which equations have a common zero?

1. $x^2-11 \cdot x+28=0$
2. $x^2-5 \cdot x-84=0$
3. $x^2-11 \cdot x+30=0$
4. $x^2-14 \cdot x+49=0$

Answer: The final answer: $x^2-11\cdot x+28=0 \land x^2-14\cdot x+49=0$
Subject: multivariable_calculus

Problem: Find the gradient of the function $f(x,y) = \frac{ \sqrt{x} + y^2 }{ x \cdot y }$.

Answer: $\nabla f(x,y)$ = $\left\langle\frac{1}{2\cdot x\cdot y\cdot\sqrt{x}}-\frac{\sqrt{x}+y^2}{y\cdot x^2},\frac{2}{x}-\frac{\sqrt{x}+y^2}{x\cdot y^2}\right\rangle$
Subject: sequences_series

Problem: Find the Fourier series of the periodic function $f(x) = \frac{ x^2 }{ 2 }$ in the interval $-2 \cdot \pi \le x < 2 \cdot \pi$ if $f(x) = f(x + 4 \cdot \pi)$.

Answer: The Fourier series is: $\frac{2\cdot\pi^2}{3}+\sum_{n=1}^\infty\left(\frac{8\cdot(-1)^n}{n^2}\cdot\cos\left(\frac{n\cdot x}{2}\right)\right)$
Subject: integral_calc

Problem: Solve the integral:
$$
\int \sqrt{\frac{ 4 \cdot \sin(4 \cdot x) }{ 9 \cdot \cos(4 \cdot x)^9 }} \, dx
$$

Answer: $\int \sqrt{\frac{ 4 \cdot \sin(4 \cdot x) }{ 9 \cdot \cos(4 \cdot x)^9 }} \, dx$ = $C+\frac{1}{9}\cdot\left(\tan(4\cdot x)\right)^{\frac{3}{2}}+\frac{1}{21}\cdot\left(\tan(4\cdot x)\right)^{\frac{7}{2}}$
Subject: multivariable_calculus

Problem: Use the method of Lagrange multipliers to find the maximum and minimum values of the function $f(x,y,z) = x^2 + y^2 + z^2$ subject to the constraint $x \cdot y \cdot z = 4$.

Answer: Minimum: $6\cdot\sqrt[3]{2}$

Maximum: None
Subject: sequences_series

Problem: Find the Fourier series of the function $\varphi(x) = 2 \cdot x$ in the interval $(0, 4 \cdot \pi)$.

Answer: The Fourier series is: $4\cdot\pi-8\cdot\sum_{n=1}^\infty\left(\frac{\sin\left(\frac{n\cdot x}{2}\right)}{n}\right)$
Subject: integral_calc

Problem: Solve the integral:
$$
\int 2 \cdot \cot(14 \cdot x)^6 \, dx
$$

Answer: $\int 2 \cdot \cot(14 \cdot x)^6 \, dx$ = $C-\frac{1}{7}\cdot\left(\frac{1}{5}\cdot\left(\cot(14\cdot x)\right)^5+\cot(14\cdot x)-\frac{1}{3}\cdot\left(\cot(14\cdot x)\right)^3-\arctan\left(\cot(14\cdot x)\right)\right)$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ 1 }{ 2 \cdot \sin\left(\frac{ x }{ 2 }\right)^6 } \, dx
$$

Answer: $\int \frac{ 1 }{ 2 \cdot \sin\left(\frac{ x }{ 2 }\right)^6 } \, dx$ = $C-\frac{1}{5}\cdot\left(\cot\left(\frac{x}{2}\right)\right)^5-\frac{2}{3}\cdot\left(\cot\left(\frac{x}{2}\right)\right)^3-\cot\left(\frac{x}{2}\right)$
Subject: algebra

Problem: Divide rational expressions and simplify it:

$$
\frac{ x^2-1 }{ 2 } \div \frac{ x-1 }{ 5 }
$$

Answer: The final answer: $\frac{5}{2}\cdot x+\frac{5}{2}$
Subject: precalculus_review

Problem: Using the table above, estimate the logarithm.

1. $\ln(16)$
2. $\ln\left(3^4\right)$
3. $\ln(2.5)$
4. $\ln\left(\sqrt{630}\right)$
5. $\ln(0.4)$

Estimate the values using the table provided.

Answer: 1. $\ln(16)$≈ $2.78$
2. $\ln\left(3^4\right)$≈ $4.4$
3. $\ln(2.5)$≈ $0.92$
4. $\ln\left(\sqrt{630}\right)$≈ $3.225$
5. $\ln(0.4)$≈ $-0.92$
Subject: precalculus_review

Problem: Which expression is equivalent to $\left(\frac{ 1 }{ \sin(x) }+\frac{ 1 }{ \tan(x) }\right)^2$?

Answer: The final answer: $T=\frac{1+\cos(x)}{1-\cos(x)}$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \sin\left(\frac{ x }{ 2 }\right)^6 \cdot \cos\left(\frac{ x }{ 2 }\right)^2 \, dx
$$

Answer: $\int \sin\left(\frac{ x }{ 2 }\right)^6 \cdot \cos\left(\frac{ x }{ 2 }\right)^2 \, dx$ = $\frac{\sin\left(\frac{x}{2}\right)^7\cdot\cos\left(\frac{x}{2}\right)}{4}+\frac{1}{8}\cdot\left(-\frac{1}{3}\cdot\sin\left(\frac{x}{2}\right)^5\cdot\cos\left(\frac{x}{2}\right)-\frac{5}{12}\cdot\sin\left(\frac{x}{2}\right)^3\cdot\cos\left(\frac{x}{2}\right)-\frac{5}{8}\cdot\sin\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{2}\right)+\frac{5}{16}\cdot x\right)+C$
Subject: precalculus_review

Problem: Find all values of $t$ that satisfy the following equation:
$$
3^{t^2} = \frac{ 1 }{ 3^t } \cdot 27^{-(1+t)}
$$

Answer: $t$ = $-3$, $-1$
Subject: algebra

Problem: Use synthetic division to find the quotient and remainder. Ensure the equation is in the form required by synthetic division:

$$
\frac{ x^3-21 \cdot x^2+147 \cdot x-343 }{ x-7 }
$$

Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor. Solve on a paper, if it is more convenient for you.

Answer: Quotient: $x^2-14\cdot x+49$ Remainder: $0$
Subject: integral_calc

Problem: Compute the integral:
$$
3 \cdot \int \frac{ \cos(2 \cdot x)^4 }{ \sin(2 \cdot x) } \, dx
$$

Answer: $3 \cdot \int \frac{ \cos(2 \cdot x)^4 }{ \sin(2 \cdot x) } \, dx$ = $\frac{3}{2}\cdot\left(C+\frac{1}{3}\cdot\left(\cos(2\cdot x)\right)^3+\cos(2\cdot x)-\frac{1}{2}\cdot\ln\left(\frac{\left|1+\cos(2\cdot x)\right|}{\left|\cos(2\cdot x)-1\right|}\right)\right)$
Subject: precalculus_review

Problem: Calculate the sum $S=\cos(a)+\cos(2 \cdot a)+\cos(3 \cdot a) + \ldots + \cos(n \cdot a)$, where $a \ne 2 \cdot \pi \cdot k$ for integers $k$.

Answer: The final answer: $S=\frac{\sin\left(\frac{n\cdot a}{2}\right)\cdot\cos\left(\frac{(n+1)\cdot a}{2}\right)}{\sin\left(\frac{a}{2}\right)}$
Subject: differential_calc

Problem: Find $\left(\left(f(a)\right)^{-1}\right)'$ for the function $f(x) = x + \sqrt{x}$ at $a = 2$.

Answer: $\left(\left(f(a)\right)^{-1}\right)'$ = $\frac{2}{3}$
Subject: sequences_series

Problem: Find the 3rd order Taylor polynomial $P_{3}(x)$ of $f(x) = \sqrt[3]{2 \cdot x + 1}$ about $a = 0$.

Answer: The final answer: $1+\frac{2}{3}\cdot x-\frac{4}{9}\cdot x^2+\frac{40}{81}\cdot x^3$
Subject: differential_calc

Problem: Find asymptotes of $y = \frac{ 3 \cdot x }{ 2 } \cdot \ln\left(e-\frac{ 1 }{ 3 \cdot x }\right)$. Submit as your final answer:

1. the equation(s) of horizontal asymptotes
2. the equation(s) of vertical asymptotes
3. the equation(s) of slant asymptotes

Answer: 1. Horizontal Asymptote(s): None
2. Vertical Asymptote(s): $x=\frac{1}{3\cdot e}$
3. Slant Asymptote(s): $y=\frac{3\cdot x}{2}-\frac{1}{2\cdot e}$
Subject: sequences_series

Problem: Using the series expansion for the function $(1+x)^m$, calculate approximately $\sqrt[3]{29}$ with an accuracy of 0.0001.

Answer: The final answer: $3.0722$
Subject: sequences_series

Problem: Find the Taylor series for $f(x) = \frac{ x }{ (2+x)^3 }$, centered at $x=-1$. Write out the sum of the first four non-zero terms, followed by dots.

Answer: The final answer: $x\cdot\left(1-3\cdot(x+1)+6\cdot(x+1)^2-10\cdot(x+1)^3+\cdots\right)$
Subject: sequences_series

Problem: Write the Taylor series for the function $f(x) = x \cdot \cos(2 \cdot x)$ at the point $x = \frac{ \pi }{ 2 }$ up to the third term (zero or non-zero).

Answer: The final answer: $-\frac{\pi}{2}-\left(x-\frac{\pi}{2}\right)+\pi\cdot\left(x-\frac{\pi}{2}\right)^2$
Subject: sequences_series

Problem: Find the sum of the series $\sum_{n=1}^\infty \frac{ x^{\frac{ n }{ 2 }+1} }{ \frac{ n }{ 2 }+1 }$. (Use differentiation of the series)

Answer: The sum of the series is $-x-2\cdot\sqrt{x}-2\cdot\ln\left(\left|\sqrt{x}-1\right|\right)$
Subject: precalculus_review

Problem: Consider the function $f(x) = \frac{ 1 }{ 2 } \cdot x^5 + 2 \cdot x$. Let $g$ denote the inverse of $f$. Find the derivative $g'(2.5)$ using the theorem $g'(c) = \frac{ 1 }{ f'\left(g(c)\right) }$.

Answer: $g'(2.5)$ = $\frac{2}{9}$
Subject: sequences_series

Problem: Compute $\int_{0}^{\frac{ 1 }{ 3 }} e^{-\frac{ x^2 }{ 3 }} \, dx$ with accuracy $0.00001$.

Answer: The final answer: $0.32926$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ \sin\left(\frac{ x }{ 2 }\right)^4 }{ \cos\left(\frac{ x }{ 2 }\right)^2 } \, dx
$$

Answer: $\int \frac{ \sin\left(\frac{ x }{ 2 }\right)^4 }{ \cos\left(\frac{ x }{ 2 }\right)^2 } \, dx$ = $\frac{2\cdot\sin\left(\frac{x}{2}\right)^3}{\cos\left(\frac{x}{2}\right)}-\frac{3}{2}\cdot x+\frac{3}{2}\cdot\sin(x)+C$
Subject: multivariable_calculus

Problem: Evaluate $\int\int\int_{E}{\left(y \cdot \ln(x)+z\right) d V}$, where $E$ is the region defined by:
$$
E = \left\{(x,y,z) | 1 \le x \le e, 0 \le y \le \ln(x), 0 \le z \le 1\right\}
$$

Answer: $I$ = $\frac{7}{2}-e$
Subject: sequences_series

Problem: Write the Taylor series for the function $f(x) = x \cdot \cos(x)$ at the point $x = \frac{ \pi }{ 2 }$ up to the third term (zero or non-zero).

Answer: The final answer: $0-\frac{\pi}{2}\cdot\left(x-\frac{\pi}{2}\right)-\left(x-\frac{\pi}{2}\right)^2$