JEE Main 2015 (Offline)Question 1
The integral $\int\frac{dx}{x^{2}\left(x^{4}+1\right)^{3/4}}$ equals :
Indefinite Integration
Solution
Concept. Take $x^4$ out of the bracket; then $t=1+x^{-4}$. Formulas used. (1) $t=1+x^{-4}\Rightarrow dt=-4x^{-5}\,dx$Step 1 - Rewrite. $$\frac{1}{x^2(x^4+1)^{3/4}}=\frac{1}{x^2\cdot x^3(1+x^{-4})^{3/4}}=\frac{x^{-5}}{(1+x^{-4})^{3/4}}.$$ Step 2 - Use (1). $$-\frac14\int t^{-3/4}\,dt=-t^{1/4}+c=-\left(\frac{x^4+1}{x^4}\right)^{1/4}+c.$$ Answer: (B)