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

$\textbf{Mountain Hiker}$

John starts to hike up a mountain at $7:00$ am and reaches the top at $7:00$ pm. He stays at the top overnight. On the next day, he starts to hike down at $7:00$ am along the same route and reaches his starting point at $7:00$ pm. His speed during the two-day hiking varies from time to time. What is the probability that there exists one spot he passes at the exactly same time during the two days?

Let function $f(x)$ satisfy:  $$\int^1_0 3f (x) dx +\int^2_1 2f (x) dx = 7$$

and $$\int^2_0 f (x) dx + \int^2_1 f (x) dx = 1$$

Find the value of $$\int^2_0 f (x) dx$$

Let $f(c)=\int_0^1\left( (x-c)^2 + c^2\right)dx$ where $c$ is a real number. Find the minimal value of $f(c)$ as $c$ varies and the maximum value of $f(\sin\theta)$ as $\theta$ varies.

Find the area of the region bounded by the curve $y=\sqrt{x}$, the line line $y=x-2$, and the $x-$ axis.

Find the number of $k$ such that the function $y=e^{kx}$ satisfies the equation $$\left(\frac{d^2y}{dx^2}+\frac{dy}{dx}\right)\left(\frac{dy}{dx}-y\right)=y\frac{dy}{dx}$$

Find the value of $c$ such that two parabolas $y=x^2+c$ and $y^2=x$ touch at a single point.