Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int \frac{1}{x^{2}\left(x^{4}+1\right)^{3 / 4}} d x$

Solution:

$I=\int \frac{1}{x^{2}\left(x^{4}+1\right)^{\frac{3}{4}}} d x$

$\Rightarrow \int \frac{1}{x^{5}\left(1+\frac{1}{x^{4}}\right)^{\frac{3}{4}}} d x$

Let $1+\frac{1}{x^{4}}=t$

$\Rightarrow-\frac{4}{x^{5}} d x=d t$

$\Rightarrow \frac{1}{x^{5}} d x=\frac{-d t}{4}$

$I=\frac{-1}{4} \int \frac{1}{t^{\frac{3}{4}}} d t$

$\Rightarrow \frac{-1}{4}\left(\frac{\mathrm{t}^{\frac{1}{4}}}{\frac{1}{4}}\right)+\mathrm{C}$

$\Rightarrow-t^{\frac{1}{4}}+c$

But $t=1+\frac{1}{x^{4}}$

$\Rightarrow-\left(1+\frac{1}{x^{4}}\right)^{\frac{1}{4}}+c$

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