Question

If \(\int \frac{dx}{x^3 (1+x^6)^{\frac{2}{3}}} = f(x) (1+ x^{6})^{\frac{1}{3}} + C\) where C is a constant of integration, then \(f(x)\) 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

\(\int \frac{dx}{x^3(1+x^6)^{\frac{2}{3}}}\)

=\(\int \frac{dx}{x^7(1+x^6)^{\frac{2}{3}}}\)
Let \(1 + \frac{1}{x^6} = t \quad \Rightarrow \quad -6x^7 \, dx = dt\)
\(∴\) \(I = -\frac{1}{6} \int \frac{dt}{t^{\frac{2}{3}}}\)
\(=\)\(-\frac{3}{6}t^{\frac{1}{3}} + C\)
\(=\)\(-\frac{1}{2}(1 + \frac{1}{x^6})^{\frac{1}{3}} + C\)
\(=\)\(-\frac{1}{2x^2}(1 + x^6)^{\frac{1}{3}} + C\)
\(∴\) \(f(x)=−\frac{1}{2x^3}\)