Question

Integrate the following :

\(\int \frac1{x^3\sqrt{(x^4 + 1)}}dx\)

Answer 1

\(\int \frac1{x^3\sqrt{(x^4 + 1)}}dx\)

\(= \int \frac 1 {x^5 \left(1 + \frac1{x^4}\right)^{\frac12}}dx\)

Put \(1 + \frac1 {x^4 } = t\)

⇒ \(\frac{-4}{x^5}dx = dt\)

⇒ \(\frac{dx}{x^5}= \frac{-dt}{4}\)

\(\therefore \int \frac{1}{x^5 \sqrt{1+\frac1{x^4}}}dx = \frac{-1}4\int\frac{dt}{\sqrt t}= \frac{-1}{4}\times 2 \sqrt t+C\)

\(= \frac{-1}{2}\sqrt{1+ \frac1{x^4}}+C\).