Question

Solve $\int \frac{x}{\sqrt{x+5}}dx$

Answer 1

Let $I=\int \frac{x}{\sqrt{x+5}}dx$

Put $\sqrt{x+5}=t$

$x=t^2-5$

$\frac{d}{dx}\sqrt{x+5}=\frac{dt}{dx}$

$\frac{dx}{2\sqrt{x+5}}=dt$

Substituting $x$ by $t$ in $I$:

$I=\int 2(t^2-5)dt$

$I=2(\frac{t^3}{3}-5t)$

$\therefore I=2t(\frac{t^2}{3}-5)$