Problem on Indefinite Integration (Calculus)

Question

How do I solve $\displaystyle\int \dfrac{dx}{x^{2} (x^{4} + 1)^{\frac{3}{4}}}$.

Answer 1

Let $$I = \int\frac{1}{x^2(x^4+1)^{\frac{3}{4}}}dx = \int\frac{1}{x^2\cdot x^{3}\left(1+x^{-4}\right)^{\frac{3}{4}}}dx$$

Now Put $(1+x^{-4}) = t\;,$ Then $\displaystyle x^{-5}dx = -\frac{1}{4}dt$

So we get $$I = -\frac{1}{4}\int t^{-\frac{3}{4}}dt = -\frac{1}{4}\cdot 4t^{\frac{1}{4}}+\mathcal{C} = -\left(1+x^{-4}\right)^{\frac{1}{4}}+\mathcal{C}$$