Compute $$\int_{-2}^{0}\frac{x^3 + 4x^2 + 7x -20}{x^2+4x+8}dx+\int_0^2\frac{2x^3-7x^2+9x-10}{x^2+4}dx$$

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 $$\sum_{n=0}^{\infty}\frac{1}{p_n(0)}$$

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 $$\int_0^2\sqrt{\frac{4-x}{x}}-\sqrt{\frac{x}{4-x}}dx$$

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

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

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

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

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

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