Integrate the following functions w.r.t. x: (5-3 x)(2-3 x)^ -1 2 - Vidyadip

Question

Integrate the following functions w.r.t. x:
$(5-3 x)(2-3 x)^{-\frac{1}{2}}$

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Answer

Let $I=\int(5-3 x)(2-3 x)^{-\frac{1}{2}} d x$Put $2-3 x=t$
$\therefore-3 d x=d t$
$\therefore d x=\frac{-d t}{3}$
Also, $x=\frac{2-t}{3}$
$ \therefore I =\int\left[5-3\left(\frac{2-t}{3}\right)\right] t^{-\frac{1}{2}} \cdot\left(\frac{-d t}{3}\right)$
$=-\frac{1}{3} \int(5-2+t) t^{-\frac{1}{2}} d t$
$ =-\frac{1}{3} \int(3+t) t^{-\frac{1}{2}} d t$
$ =-\frac{1}{3} \int\left(3 t^{-\frac{1}{2}}+t^{\frac{1}{2}}\right) d t$
$=-\frac{3}{3} \int t^{-\frac{1}{2}} d t-\frac{1}{3} \int t^{\frac{1}{2}} d t$
$=-\frac{t^{\frac{1}{2}}}{(1 / 2)}-\frac{1}{3} \cdot \frac{t^{\frac{3}{2}}}{(3 / 2)}+c$
$=-2 \sqrt{2-3 x}-\frac{2}{9}(2-3 x)^{\frac{3}{2}}+c .$

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