∫(xdx)/((x^(2)+4)^(3) )

Question

∫
`(xdx)/((x^(2)+4)^(3) )`

Answer 1

Substitute
`u=x^2+4` and
`du=2x\,dx`. Then the integral transforms to

`\displaystyle \int (x\,dx)/((x^2+4)^3) = \frac12 \int (du)/(u^3)`

Apply the power rule.

`\displaystyle \int (du)/(u^3) = -\frac1{2u^2} + C`

Now put the result back in terms of
`x`.

`\displaystyle \int (x\,dx)/((x^2+4)^3) = \frac12 \left(-\frac1{2u^2} + C\right) = -\frac1{4u^2} + C = \boxed{-\frac1{4(x^2+4)^2} + C}`