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Let $f(x)=\int_1^x\frac{\ln{x}}{1+x}dx$ for $x > 0$. Find $f(2)+f(\frac{1}{2})$.


Compute $$\lim_{x\to 0}\frac{\int_0^x\sin(xt)^2dt}{x^5}$$


Compute

$$\int_0^{\infty}\frac{x^2}{1+x^4}dx$$


Find the value of $$\displaystyle\lim_{n\to\infty}\sum_{k=1}^{n}\frac{n+k}{n^2 + k}$$

Evaluate $\displaystyle\lim_{n\to\infty}S_n$ where

$$S_n = 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdots + (-1)^{n-1}\frac{1}{n}$$


Determine the values of $\alpha$ and $\beta$ such that 

$$\lim_{n\to\infty}\frac{n^{\alpha}}{n^{\beta}-(n-1)^{\beta}}=2020$$


Evaluate

$$\int_0^1 x\arcsin{x}d{x}$$


Evaluate

$$\int_0^1 \sqrt{1-x^2} d{x}$$


Compute

$$I= \iiint \limits_S \frac{dx dy dz}{(1+x+y+z)^2}$$

where $S=\{x\ge 0, y\ge 0, z\ge 0, x+y+z\le 1\}$.


Compute $$\int \ln{x} dx$$


Which one of the numbers below is larger?

$$\int_0^{\pi} e^{\sin^2x}dx\qquad\text{and}\qquad \frac{3\pi}{2}$$


Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a periodic continuous function of period $T > 0$, that is $f(x+T)=f(x)$ holds for any $x\in\mathbb{R}$. Show that

$$\lim_{x\to\infty}\frac{1}{x}\int_0^xf(t)dt=\frac{1}{T}\int_0^Tf(t)dt$$


Prove the absolute convergence testing rule using the comparison testing rule. That is, if a series $\{|a_n|\}$ converges, then the series $\{a_n\}$ must be convergent.


For what pairs $(a, b)$ of positive real numbers does the the following improper integral converge?

$$\int_b^{\infty}\left(\sqrt{\sqrt{x+a}-\sqrt{x}}-\sqrt{\sqrt{x}-\sqrt{x-b}}\right)dx$$


Show that the function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ given by

$$f(x,y)=x^4+6x^2y^2 + y^4 -\frac{9}{4}x-\frac{7}{4}$$

achieves its minimal value, and determine all the points $(x, y)\in\mathbb{R}^2$ at which it is achieved.


For $n=1, 2,\dots$, let $x_n=\displaystyle\sum_{k=n+1}^{9n}\frac{k}{9n^2 + k^2}$. Find the value of $\displaystyle\lim_{n\to\infty}x_n$.


Let $s\in\mathbb{R}$. Prove that

$$\sum_{n\ge 1}(n^{\frac{1}{n^s}}-1)$$

converges if and only if $s > 1$.


A right circular cone $\mathbb{C}$ has altitude $40$ and a circular base of radius $30$ inches. A sphere $\mathbb{S}$ is inscribed in $\mathbb{C}$. Compute the volume of the region inside $\mathbb{C}$ which is above $\mathbb{S}$.


Let $$S_n=\sum_{k=1}^{2n}\frac{1}{n+k}=\frac{1}{n+1}+\frac{1}{n+2}+\cdots + \frac{1}{3n}$$

Does $\displaystyle\lim_{n\to\infty}S_n$ exist? If so, find its value. If not, prove the claim.


Show that $1-\cos{x} < x^2$ holds for all $x > 0$.


Determine whether the following series converge? $$\sum_{n=1}^{\infty}\left(1-\cos{\frac{\pi}{n}}\right)$$


Note that the point $(2, 1)$ is always on the curve $x^4 + ky^4 = 16+k$ regardless of the value of $k$. If for a particular non-zero value of $k$, $y'(2)=y''(2)$ along this curve. Find this $k$.


According to Newton’s law of cooling, the rate at which a cup of coffee cools is proportional to the difference between its temperature and that of the room it is in. A certain cup of coffee cools from $164^{\circ}$ to $140^{\circ}$ (all temperatures Fahrenheit) in five minutes, and then from $140^{\circ}$ to $122^{\circ}$ in the next five minutes. What is the temperature of the room?


It is well-known that the solution to the Fibonacci sequence is

$$F_n=\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right)$$

Show that

$$\lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\frac{1+\sqrt{5}}{2}$$


Compute $$\lim_{n\to\infty}\left(\sqrt{n+1}-\sqrt{n}\right)$$