Evaluate the indefinite integral:
$$\int\frac{x^2+3x+3}{(x+1)^3}e^{-x}\sin x\ dx$$
Search found 14 matches
- Sat Jan 16, 2016 3:51 pm
- Forum: Calculus
- Topic: $\int\frac{x^2+3x+3}{(x+1)^3}e^{-x}\sin x\ dx$
- Replies: 1
- Views: 2119
Re: Integral
\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} \...
- Sat Dec 19, 2015 4:14 pm
- Forum: Calculus
- Topic: $\int_0^\infty \frac{\sin^2 (\tan x)}{x^2}\, dx$
- Replies: 3
- Views: 3574
$\int_0^\infty \frac{\sin^2 (\tan x)}{x^2}\, dx$
Evaluate the integral:
$$\int_0^\infty \frac{\sin^2 (\tan x)}{x^2}\, dx$$
$$\int_0^\infty \frac{\sin^2 (\tan x)}{x^2}\, dx$$
- Fri Dec 11, 2015 7:57 pm
- Forum: Calculus
- Topic: Integral with fractional part
- Replies: 2
- Views: 2654
Integral with fractional part
Let $n \in \mathbb{N} \setminus \{1, 2\}$. Prove that:
$$\int_1^\infty \frac{\{x\}}{x^n}\, dx = \frac{1}{n-2} - \frac{\zeta(n-1)}{n-1}$$
$$\int_1^\infty \frac{\{x\}}{x^n}\, dx = \frac{1}{n-2} - \frac{\zeta(n-1)}{n-1}$$