QJEEM15Ma3a66: JEE Main 2015 Mathematics - Class 12 JEEM15Ma3a66 Solution: The integral equals :

Question

Verified Solution • 2025–26 Session moderate

The integral

\int {{{dx} \over {{x^2}{{\left( {{x^4} + 1} \right)}^{3/4}}}}}

equals :

Accepted Answer

1 = \int {{{dx} \over {{x^2}{{\left( {{x^4} + 1} \right)}^{3/4}}}}}

= \int {{{dx} \over {{x^3}{{\left( {1 + {x^{ - 4}}} \right)}^{3/4}}}}}

Let

{x^{ - 4}} = y

\Rightarrow - 4{x^{ - 3}}\,dx = dy

\Rightarrow dx = {{ - 1} \over 4}{x^3}dy

\therefore

I = {{ - 1} \over 4}\int {{{{x^3}dy} \over {{x^3}{{\left( {1 + y} \right)}^{3/4}}}}}

= {{ - 1} \over 4}\int {{{dy} \over {{{\left( {1 + y} \right)}^{3/4}}}}}

= {{ - 1} \over 4} \times 4{\left( {1 + y} \right)^{1/4}}

= - 1{\left( {1 + {x^{ - 4}}} \right)^{1/4}} + C

= - {\left( {{{{x^4} + 1} \over {{x^4}}}} \right)^{1/4}} + C

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