Question

Solve :
$I=\int \frac{x+9}{x^2+5}dx$

Answer 1

$I=\int \frac{x^2+9}{x^2+5}dx$

$I=\int \frac{x}{x^2+5}dx+\int \frac{9}{x^2+5}dx$

$\left(A\right)$ $\left(B\right)$

Consider 'A' Consider 'B'

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

Let $x^2+5=p$ $\Rightarrow 9\int \frac{1}{x^2+(\sqrt{5}}{)}^{2}dx$

$2xdx=dp$ $\Rightarrow \frac{9}{\sqrt{5}}ta{n}^{-1}\left(\frac{x}{\sqrt{5}}\right)+d$

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

$\Rightarrow \frac{1}{2}\int \frac{dp}{p}$ $(\therefore \int \frac{1}{x^2+a^2}=ata{n}^{-1}(\frac{x}{a})$

$\frac{1}{2}|n|p|+c$...(1)

$I=A+B$

$I=\frac{1}{2}|n|x^2+5|+\frac{9}{\sqrt{5}}{\tan }^{-1}\left(\frac{x}{\sqrt{5}}\right)+k$

When $k=constant$