Compute $$\int_0^{\pi}\frac{2x\sin{x}}{3+\cos^2x}dx$$

Given that the value $\ln(2)$ is not the root of any polynomial with rational coefficients. For any nonnegative integer $n$, let $p_n(x)$ be the unique polynomial with integer coefficients such that $$p_n(\ln(2)) =\int_1^2 (ln(x))^n dx$$

Compute the value of the $$\sum_{n=0}^{\infty}\frac{1}{p_n(0)}$$

There is a unique positive real number $a$ such that the tangent line to $y = x^2 + 1$ at $x = a$ goes through the origin. Compute $a$.

Moor has $\$1000$, and he is playing a gambling game. He gets to pick a number k between $0$ and $1$ (inclusive). A fair coin is then flipped. If the coin comes up heads, Moor is given $5000k$ additional dollars. Otherwise, Moor loses $1000k$ dollars. Moor’s happiness is equal to the log of the amount of money that he has after this gambling game. Find the value of k that Moor should select to maximize his expected happiness.

The set of points $(x, y)$ in the plane satisfying $x^{2/5} + |y| = 1$ form a curve enclosing a region. Compute the area of this region.

Compute the value of $$\int_0^2\sqrt{\frac{4-x}{x}}-\sqrt{\frac{x}{4-x}}dx$$

Compute $$\lim_{x\to\infty}\left[x-x^2\ln\left(\frac{1+x}{x}\right)\right]$$

For a given $x > 0$, let $a_n$ be the sequence defined by $a_1=x$ for $n = 1$ and $a_n = x^{a_{n−1}}$ for $n\ge 2$. Find the largest $x$ for which the limit $\displaystyle\lim_{n\to\infty} a_n$ converges.

Find the derivative of $x^x$.

Let a differentiable function $f(x)$ satisfy $$f(x)\cos{x} + 2\int_0^xf(t)\sin{t}dt = x+1$$

Find $f(x)$.

Find the value of $\displaystyle\lim_{n\to\infty}\sin^2\left(\pi\sqrt{n^2+n}\right)$.

Find the value of $$I=\int\frac{e^{-\sin{x}}\sin(2x)}{(1-\sin{x})^2}dx$$

Compute the value of $$\lim_{x\to\pi}\frac{\ln(2+\cos{x})}{\left(3^{\sin{x}}-1\right)^2}$$

Compute the value of $$\lim_{n\to\infty}n^2\left(1-\cos\frac{\pi}{n}\right)$$

Let $$f(x)=\left\{\begin{array}{ll} \cos{x} &, x\in[-\frac{\pi}{2}, 0)\\e^x&,x\in[0,1] \end{array}\right.$$

Compute $\displaystyle\int_{-\frac{\pi}{2}}^{1}f(x)dx$.

Determine whether or not these two series converge: $$(A)\ \ \sum_{n=1}^{\infty}\sin\left(\frac{\cos{n}}{n^2}\right)\qquad (B)\ \  \sum_{n=1}^{\infty}\cos\left(\frac{\sin{n}}{n^2}\right)$$

The equation $x^y=y^x$ describes a curve in the first quadrant of the plane containing the point $P=(4, 2)$. Compute the slope of the line that is tangent to this curve at $P$.

Compute $$\int\frac{1}{\sin{x}}d{x}$$

Compute $$\int_0^{\frac{\pi}{4}}\frac{1}{\sin{x}+\cos{x}}d{x}$$

If water is poured into a right cone whose height is $H$ cm and base's radius is $R$ cm at a speed of $A$ $cm^3$ per second, what is the speed the water is rising when the depth of water is half of the cone's height?

Let $f(x)=x^2\cos(ax)$ where $a$ is a constant. Find the $50^{th}$ order derivative of $f(x)$, i.e. $f^{(50)}(x)$.

Estimate the value of $\sqrt[4]{10018}$.

The three segments marked with lengths are perpendicular to each other. Find the area of the outside square.

Compute $$\int\frac{1}{ax+b}d{x}$$