Evaluate: x-2 x+5 d x - Vidyadip

Question

Evaluate:

$\int \frac{x-2}{\sqrt{x+5}} \cdot d x$

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Answer

$\begin{aligned} & \int \frac{x-2}{\sqrt{x+5}} d x=\int \frac{(x+5)-7}{\sqrt{x+5}} d x \\ = & \int\left(\frac{x+5}{\sqrt{x+5}}-\frac{7}{\sqrt{x+5}}\right) d x \\ = & \int(x+5)^{\frac{1}{2}} d x-7 \int(x+5)^{-\frac{1}{2}} d x \\ = & \frac{(x+5)^{\frac{3}{2}}}{(3 / 2)}-\frac{7(x+5)^{\frac{1}{2}}}{(1 / 2)}+c\end{aligned}$

$=\frac{2}{3}(x+5)^{\frac{3}{2}}-14 \sqrt{x+5}+c$.

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