How to evaluate this integral for this type?

Question

What is the skill to integrate this type of integral

$\displaystyle\int \frac{4x^n}{x^2+9} dx$ for $n$ is constant

How to use your general method to work out an example?

Answer 1

Let $x=3\tan t\;\Rightarrow\;dx=3\sec^2t\ dt$, then $$ \int\frac{x^n}{x^2+9}\ dx=3^{n-1}\int\tan^n t\ dt. $$ Now, use integration by the reduction formula. $$ \begin{align} \int \tan^n t\,dt &= \int \tan^{n-2}t\,(\sec^2 t - 1)\,dt \\ &= \int \tan^{n-2}t\,\sec^2 t\,dt - \int \tan^{n-2}t\,dt \\ &= \int u^{n-2}\,\,du - \int \tan^{n-2}t\,dt \\ &= \frac{1}{n-1}u^{n-1} - \int \tan^{n-2}t\,dt \\ &= \frac{\tan^{n-1} t}{n-1} - \int \tan^{n-2}t\,dt, \end{align} $$ where $n\in\mathbb{Z}_+$ and $n\neq1$.