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

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.