Question

Evaluate: $\int \frac{x+\sqrt{x+1}}{x+2}dx$

Answer 1

Consider, $I=\int \frac{x+\sqrt{x+1}}{x+2}dx$

$=\int \frac{x}{x+2}+\frac{\sqrt{x+1}}{x+2}dx$

$=\int \frac{x+2}{x+2}-\frac{2}{x+2}dx+\int \frac{\sqrt{x+1}}{x+2}dx$

$=x-2\log (x+2)+\int \frac{\sqrt{x+1}}{x+2}dx$

put $\sqrt{x+1}=t$ thus $\int \frac{2t^2}{t^2+t}dt$

$=2\int -\frac{1}{t^2+1}dt+2\int 1dt=2({\tan }^{-1}t+t)$

$\Rightarrow \int \frac{x+\sqrt{x+1}}{x+2}dx=x2\log (x+2)+2({\tan }^{-1}\sqrt{x+1}+\sqrt{x+1})+c$