Compute $$\int\frac{1}{x^2-a^x}d{x}$$

Compute $$\int\frac{1}{\sqrt{a^2-x^2}}d{x}$$

Compute $$\int\frac{x}{1+x^2}dx$$

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

Compute $$\int\sin^5{x}dx$$

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

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

Compute $$\int\frac{1}{\sqrt{x^2+1}}dx$$

Compute $$\int\frac{1}{\sqrt{x^2+4x+5}}dx$$

Compute $$\int x^3\ln{x}d{x}$$

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

Evaluate $$\int e^{ax}\cos(bx)d{x}\quad\text{and}\quad\int e^{ax}\sin(bx)d{x}$$

Evaluate $$\int x^2e^xd{x}$$

Evaluate $$\int x^2\sin{x}dx$$

Compute $$\int\sqrt{x^2+a^2}dx$$

Evaluate $$\int\frac{1}{\sqrt{x^2 + a^2}}dx$$

Compute $$\int\frac{x+1}{x^2+x+1}dx$$

Evaluate $$\int\frac{5x+6}{x^2+3x+1}dx$$

Evaluate $$\int\frac{5x+6}{(x^2+x+2)^2}dx$$

Evaluate $$\int\frac{x}{(x^2+1)(x-1)}dx$$

$\textbf{Two Doormen}$

Two doormen are guarding two rooms. One room contains tons of gold and the other is empty. Among these two doormen, one is honest who always tells the truth and the other is a liar who always gives false answers. While they know each other well, you do not know who is honest and who is not. If you are given just one chance to ask one question to one of them, what can you do in order to find out which room contains the gold?

$\textbf{Toggler's Problem}$

Among a group of $100$ people, only one is a truth teller and the rest $99$ are togglers. A truth teller always tells the truth. A toggler will tell the truth and a lie in an alternating fashion. That is, after he or she tells the truth the first time, this person will tell a lie next time. However, if his or her first answer is false, then the next answer will be true. It is unknown whether a toggler's first answer is the truth or a lie.

If all these people know who is the truth teller, how many questions do you need to ask in order to identify the truth teller?

$\textbf{Connect the Lights}$

You are in a control room which has three switches. Each switch controls one of three lights in another room. Once you leave the control room, you can not touch the switches again. How can you figure out which switch controls which light?

$\textbf{What Bear}$

Joe leaves his campsite and hikes south for $3$ miles. He then turns east and hikes for $3$ miles. Finally he turns north and hikes for $3$ miles. At this moment he sees a bear inside his tent eating his food! What color is the bear?

$\textbf{Unique Number}$

What makes the number $8549176320$ unique?