# Evaluate:

Question:

Evaluate: $\int \frac{\mathrm{x}}{\sqrt{\mathrm{x}+4}} \mathrm{dx}$

Solution:

In these questions, little manipulation makes the questions easier to solve

Add and subtract 4 from the numerator

$\Rightarrow \int \frac{x+4-4}{\sqrt{x+4}} d x$

$\Rightarrow \int \frac{x+4-4}{\sqrt{x+4}} d x$

$\Rightarrow \int \frac{x+4}{\sqrt{x+4}} d x-\int \frac{4}{\sqrt{x+4}} d x$

$\Rightarrow\left(\int \sqrt{\mathrm{x}+4} \mathrm{dx}-4 \int(\mathrm{x}+4)^{\frac{-1}{2}} \mathrm{dx}\right)$

$\Rightarrow \frac{(x+4)^{\frac{3}{2}}}{\frac{3}{2}}-4 \times \frac{(x+4)^{\frac{1}{2}}}{\frac{1}{2}}+c$

$\Rightarrow \frac{2(x+4)^{\frac{3}{2}}}{3}-8(x+4)^{\frac{1}{2}}+c$