Integrate the following functions w.r.t. x: (2 x+1) x+2 - Vidyadip

Question

Integrate the following functions w.r.t. x:
$(2 x+1) \sqrt{x+2}$

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Answer

Let $I=\int(2 x+1) \sqrt{x+2} d x$Put $x+2=t \quad \therefore d x=d t$
Also, $x=t-2$
$ \therefore 2 x+1=2(t-2)+1=2 t-3$
$\therefore I=\int(2 t-3) \sqrt{t} d t$
$=\int\left(2 t^{\frac{3}{2}}-3 t^{\frac{1}{2}}\right) d t$
$=2 \int t^{\frac{3}{2}} d t-3 \int t^{\frac{1}{2}} d t$
$=2 \cdot \frac{t^{\frac{5}{2}}}{\left(\frac{5}{2}\right)}-3 \cdot \frac{t^{\frac{3}{2}}}{\left(\frac{3}{2}\right)}+c$
$=\frac{4}{5}(x+2)^{\frac{5}{2}}-2(x+2)^{\frac{3}{2}}+c$.

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