Question

Integrate `x/(x^2+2)^2` ?

Answer 1

`-1/(2(x^2+2))+C`

Explanation:

We use the substitution rule. We define a new variable `u` as `u=x^2+2`. This implies that `du=2x\ dx`, or `dx=(du)/(2x)`.

We substitute these values in our original problem `int\ x/(x^2+2)^2\ dx` to get `int\ (x\ du)/(2xu^2)`. Canceling out the `x`'s, we get `int\ (du)/(2u^2)`.

This becomes a simple matter of integrating `1/2u^-2` with respect to `u`. Using the constant rule and power rule in integration, we obtain `-1*1/2u^-1+C=-1/(2u)+C`.

However, we want our answer in terms of `x`. Remember that we first defined `u` as `x^2+2`. We just need to substitute this back to get `-1/(2(x^2+2))+C`.