Question:
The integral $\int \frac{1}{\sqrt[4]{(x-1)^{3}(x+2)^{5}}} \mathrm{dx}$ is equal to :
(where $C$ is a constant of integration)
Correct Option: , 3
Solution:
$\int \frac{d x}{(x-1)^{3 / 4}(x+2)^{5 / 4}}$
$=\int \frac{d x}{\left(\frac{x+2}{x-1}\right)^{5 / 4} \cdot(x-1)^{2}}$
put $\frac{x+2}{x-1}=t$
$=-\frac{1}{3} \int \frac{\mathrm{dt}}{\mathrm{t}^{5 / 4}}$
$=\frac{4}{3} \cdot \frac{1}{\mathrm{t}^{1 / 4}}+\mathrm{C}$
$=\frac{4}{3}\left(\frac{x-1}{x+2}\right)^{1 / 4}+C$
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