Question:
Evaluate: $\int \frac{\mathrm{x}-1}{\sqrt{\mathrm{x}+4}} \mathrm{dx}$
Solution:
In these questions, little manipulation makes the questions easier to solve
Add and subtract 5 from the numerator
$\Rightarrow \int \frac{x+5-5-1}{\sqrt{x+4}} d x$
$\Rightarrow \int \frac{x+4-5}{\sqrt{x+4}} d x$
$\Rightarrow \int \frac{x+4}{\sqrt{x+4}} d x-\int \frac{5}{\sqrt{x+4}} d x$
$\Rightarrow\left(\int \sqrt{x+4} d x-5 \int(x+4)^{\frac{-1}{2}} d x\right)$
$\Rightarrow \frac{(x+4)^{\frac{3}{2}}}{\frac{3}{2}}-5 \times \frac{(x+4)^{\frac{1}{2}}}{\frac{1}{2}}+c$
$\Rightarrow \frac{2(x+4)^{\frac{3}{2}}}{3}-10(x+4)^{\frac{1}{2}}+c$
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