Evaluate $\int_0^\infty \frac x {(x+1)^3}$

Question

I feel like this should be easy, but I can't remember the technique that I should use to solve this.

How does one solve an integral like this:

$$\int_0^\infty \frac x {(x+1)^3}$$

Answer 1

Try this $${x\over (x +1)^3} = {x + 1\over(x + 1)^3} - {1\over(x+1)^3}. $$

Answer 2

$$ \int_0^\infty \frac{x}{(x+1)^3} \mathrm{d} x =\int_1^\infty \frac{x-1}{x^3} \mathrm{d} x \\ =\left[-\frac1{x} + \frac12 \frac1{x^2}\right]_1^{\infty}=\frac12 $$

Answer 3

\begin{align*} & \int_0^\infty \frac{x}{(x+1)^3} \mathrm{d} x \\ & = \int_0^\infty \frac{x + 1 -1}{(x+1)^3} \mathrm{d} x \\ & = \int_0^\infty \left( \frac{1}{(x+1)^2} - \frac{1}{(x+1)^3} \right) \mathrm{d} x \\ \end{align*}

Now use the substitution $u = x+1$.