The integral \(\int\displaystyle\frac{dx}{x^2\left(x^4+1\right)^{\frac{3}{4}}}\) equals ________.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(-\left(\displaystyle\frac{x^4+1}{x^4}\right)^{\frac{1}{4}}+C\)
Take \(x^4\) out of the bracket: \(\left(x^4+1\right)^{3/4} = x^3\left(1+x^{-4}\right)^{3/4}\) (for x > 0). Then
\( \displaystyle I = \int\frac{dx}{x^5\left(1+x^{-4}\right)^{3/4}} \)
Use the substitution \(t = 1 + x^{-4}\):
\( \displaystyle dt = -4x^{-5}\,dx,\quad \frac{dx}{x^5} = -\frac{dt}{4} \)
\( \displaystyle I = -\frac{1}{4}\int t^{-3/4}\,dt \)
\( \displaystyle = -\frac{1}{4}\times 4\,t^{1/4} + C = -t^{1/4} + C \)
Putting back \(t = 1 + \frac{1}{x^4} = \frac{x^4+1}{x^4}\):
\( \displaystyle I = -\left(\frac{x^4+1}{x^4}\right)^{\frac{1}{4}} + C \)
Why the other options are incorrect
Option (1) \(\left(\displaystyle\frac{x^4+1}{x^4}\right)^{\frac{1}{4}}+C\) — Incorrect. The sign is wrong: the substitution gives \(dt = -4x^{-5}\,dx\), which brings in a minus sign.
Option (2) \(\left(x^4+1\right)^{\frac{1}{4}}+C\) — Incorrect. Its derivative is \(x^3\left(x^4+1\right)^{-3/4}\), not the given integrand.
Option (3) \(-\left(x^4+1\right)^{\frac{1}{4}}+C\) — Incorrect. Its derivative is \(-x^3\left(x^4+1\right)^{-3/4}\), not the given integrand.
NCERT (2026-27 edition):
Integrals
