The value of ∫ dfracdxx^2(x^4+1)^3/4 is — JEE Main 2015 Mathematics PYQ

JEE Main 2015

The value of

\int \dfrac{dx}{x^2(x^4+1)^{3/4}}

A.
$\left(\dfrac{x^4+1}{x^4}\right)^{\frac{1}{4}} + c$
B.
$(x^4+1)^{\frac{1}{4}} + c$
C.
$-(x^4+1)^{\frac{1}{4}} + c$
D.
$-\left(\dfrac{x^4+1}{x^4}\right)^{\frac{1}{4}} + c$

Show answer and solution

Correct option: D

Solution

Factor x⁴ from (x⁴+1)^(3/4): write (x⁴+1)^(3/4) = x³(1 + 1/x⁴)^(3/4).

The integral becomes ∫ dx / [x² · x³ · (1+x⁻⁴)^(3/4)] = ∫ x⁻⁵ (1+x⁻⁴)^(-3/4) dx.

Substitute t = 1 + x⁻⁴, so dt = -4x⁻⁵ dx, giving dx·x⁻⁵ = -dt/4.

Integral = -¼ ∫ t^(-3/4) dt = -¼ · 4t^(1/4) + c = -t^(1/4) + c = -(1 + 1/x⁴)^(1/4) + c = -((x⁴+1)/x⁴)^(1/4) + c.

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JEE Main 2015 Mathematics question on Indefinite Integration