Question

Evaluate :
$\int \frac{x^5}{x^2+9}dx$

Answer 1

Consider the given integral.

$I=\int \frac{x^5}{x^2+4}dx$

Let $t=x^2+4$

$\frac{dt}{dx}=2x+0$

$\frac{dt}{2}=xdx$

Therefore,

$I=\frac{1}{2}\int \frac{{(t-4)}^2}{t}dt$

$I=\frac{1}{2}\int \frac{t^2+16-8t}{t}dt$

$I=\frac{1}{2}\int (t+\frac{16}{t}-8)dt$

$I=\frac{1}{2}[\frac{t^2}{2}+16\log \left(t\right)-8t]+C$

On putting the value of $t$, we get

$I=\frac{1}{2}[\frac{{(x^2+4)}^2}{2}+16\log (x^2+4)-8(x^2+4)]+C$

Hence, this is the answer.