Question:
Evaluate the following integrals:
$\int \frac{(1+\sqrt{x})^{2}}{\sqrt{x}} d x$
Solution:
Assume $1+\sqrt{x}=t$
$\Rightarrow d(1+\sqrt{x})=d t$
$\Rightarrow \frac{1}{2 \sqrt{x}} \mathrm{dx}=\mathrm{dt}$
$\Rightarrow \frac{1}{\sqrt{x}} \mathrm{dx}=2 \mathrm{dt}$
$\therefore$ Substituting $t$ and $d t$ in the given equation we get
$\Rightarrow \int 2 t^{2} \cdot d t$
$\Rightarrow 2 \int t^{2} \cdot d t$
$\Rightarrow \frac{2 t^{3}}{3}+c$
But $1+\sqrt{x}=t$
$\Rightarrow \frac{2(1+\sqrt{x})^{3}}{3}+c$
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.