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  4. ∫ (1/x2(x4+1)3 / 4) d x is equal to

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Q. $\int \frac{1}{x^{2}\left(x^{4}+1\right)^{3 / 4}} d x$ is equal to

VITEEEVITEEE 2016

Solution:

$I=\int \frac{1}{x^{2}\left(x^{4}+1\right)^{3 / 4}} d x$
$=\int \frac{1}{x^{2}\left[x^{4}\left(1+\frac{1}{x^{4}}\right)\right]^{3 / 4}} d x$
$=\int \frac{1}{x^{2} \cdot x^{3}\left(1+\frac{1}{x^{4}}\right)^{3 / 4}} d x$
$I=\int \frac{1}{x^{5}\left(1+\frac{1}{x^{4}}\right)^{3 / 4}} d x$
Put $1+\frac{1}{x^{4}}=t$
$\frac{-4}{x^{5}} d x=d t$
$I=\int \frac{-d t}{4 t^{3 / 4}}$
$=-\frac{1}{4} \int t^{-3 / 4} d t$
$I=-\frac{1}{4}\left[\frac{t^{-\frac{3}{4}+1}}{\frac{-3}{4}+1}\right]+C$
$I=\frac{-1}{4} \frac{t^{1 / 4}}{1 / 4}+C$
$I=-\left(1+\frac{1}{\lambda^{4}}\right)^{1 / 4}+C$