Question

If $\int \frac{dx}{x^{3}\left(1+x^{6}\right)^{\frac{2}{3}}}=f \left(x\right)\left(1+x ^{6}\right)^{\frac{1}{3}}+C$, where C is a constant of integration, then the function $f \left(x\right)$ is equal to-

• A. $- \frac{1}{6x^{3}}$
• B. $ \frac{3}{x^{2}}$
• C. $- \frac{1}{2x^{2}}$
• D. $- \frac{1}{2x^{3}}$

Answer 1

The Correct Option is D

To solve this integral problem, we need to find the function \( f(x) \) from the given expression:

\[ \int \frac{dx}{x^{3}\left(1+x^{6}\right)^{\frac{2}{3}}}=f \left(x\right)\left(1+x ^{6}\right)^{\frac{1}{3}}+C \]

We aim to simplify the left-hand side such that it matches with the form \( f(x)(1+x^6)^{1/3} + C \).

1. Let us perform substitution:
   • Let \( t = \left(1 + x^6\right)^{1/3} \)
   • Then, differentiate both sides with respect to \( x \): \[\frac{d}{dx} \left( \left(1 + x^6\right)^{1/3} \right) = \frac{1}{3}\left(1+x^6\right)^{-\frac{2}{3}} \cdot 6x^5 = 2x^5 \left(1+x^6\right)^{-\frac{2}{3}}\] \
   • This implies, \[ dt = 2x^5 \left(1+x^6\right)^{-\frac{2}{3}} \, dx \] \
2. Rewriting the original integral:

   With substitution \( t \), the integral becomes: \[\int \frac{dx}{x^{3}\left(1+x^{6}\right)^{\frac{2}{3}}} = \int \frac{dt}{2x^5(1+x^6)^{-\frac{2}{3}}}\] \

   This simplifies to:

   \[\int \frac{1}{2x^5} \, dt = \int -\frac{1}{2} \frac{dt}{x^3}\] \
3. We can now solve the integral:
   • Recognizing this form, using partial fractions:

     \[ \int -\frac{1}{2x^3} \, dt = -\frac{1}{2x^3}\] + C \

4. So, comparing with the given expression:

   The final function \( f(x) \) is:

   \[f(x) = -\frac{1}{2x^3}\] \

Conclusion: The function \( f(x) \) is equal to \[-\frac{1}{2x^3}\] \, matching the correct answer option.