Indefinite Integration
Integration by Substitution
Grade 12
Question:
<p>[JEE Main 2021] \(\displaystyle\int\frac{dx}{x^2(x^4+1)^{3/4}}\) equals (where \(C\) is constant)</p>
<li>\(-(1+x^{-4})^{1/4}+C\)</li>
<li>\(\dfrac{(1+x^4)^{1/4}}{x}+C\)</li>
<li>\(-(1+x^{-4})^{1/4}+C\)</li>
<li>\(\dfrac{(1+x^{-4})^{1/4}}{x}+C\)</li>
Step-by-Step Solution
Key Concept: Factor x^4 from (x^4+1)^(3/4) = x^3(1+x⁻^4)^(3/4). Cancel x^3 from denominator. Substitute t=1+x⁻^4.
<p>$(x^4+1)^{3/4}=x^3(1+x^{-4})^{3/4}$.</p>
<p>$$\int\frac{dx}{x^2\cdot x^3(1+x^{-4})^{3/4}} = \int\frac{x^{-5}}{(1+x^{-4})^{3/4}}\,dx$$</p>
<p>Let $t=1+x^{-4}\Rightarrow dt=-4x^{-5}\,dx\Rightarrow x^{-5}\,dx=-dt/4$:</p>
<p>$$= -\frac{1}{4}\int t^{-3/4}\,dt = -\frac{1}{4}\cdot\frac{t^{1/4}}{1/4}+C = -(1+x^{-4})^{1/4}+C$$</p>
<p>Answer: <strong>(A)</strong></p>
Correct Answer: A