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Evaluate the indefinite integral:


$$\int\frac{x^2+3x+3}{(x+1)^3}e^{-x}\sin x\ dx$$

\begin{aligned} \int \frac{1-a^2}{1-2a \cos x + a^2}\, dx &=(1-a^2)\int \frac{1}{\left ( a^2+1 \right ) \left ( \cos^2 \frac{x}{2}+ \sin^2 \frac{x}{2} \right ) - 2a \left ( \cos^2 \frac{x}{2} - \sin^2 \frac{x}{2} \right )} \, dx \\ &= \left ( 1-a^2 \right ) \int \frac{1}{\cos^2 \frac{x}{2} \...

Evaluate the integral:

$$\int_0^\infty \frac{\sin^2 (\tan x)}{x^2}\, dx$$

Let $n \in \mathbb{N} \setminus \{1, 2\}$. Prove that:

$$\int_1^\infty \frac{\{x\}}{x^n}\, dx = \frac{1}{n-2} - \frac{\zeta(n-1)}{n-1}$$