Indefinite Integration
Integration of Rational Functions
Grade 12
Question:
<p>[JEE Main 2019] \(\displaystyle\int\frac{dx}{x(x^n+1)}\) equals</p>
<li>\(\dfrac{1}{n}\ln\!\dfrac{x^n}{x^n+1}+C\)</li>
<li>\(\dfrac{1}{n}\ln\!\dfrac{x^n+1}{x^n}+C\)</li>
<li>\(\ln\!\left|\dfrac{x}{x^n+1}\right|+C\)</li>
<li>\(\dfrac{1}{n}\tan^{-1}(x^n)+C\)</li>
Step-by-Step Solution
Key Concept: Multiply top and bottom by xⁿ⁻^1. Let t=xⁿ, dt=n \cdot xⁿ⁻^1 dx. Integral becomes (1/n)\intdt/(t(t+1)).
<p>Multiply by $\dfrac{x^{n-1}}{x^{n-1}}$:</p>
<p>$$\int\frac{x^{n-1}}{x^n(x^n+1)}\,dx$$</p>
<p>Let $t=x^n\Rightarrow dt=nx^{n-1}dx$:</p>
<p>$$= \frac{1}{n}\int\frac{dt}{t(t+1)} = \frac{1}{n}\int\left(\frac{1}{t}-\frac{1}{t+1}\right)dt = \frac{1}{n}\ln\frac{t}{t+1}+C = \frac{1}{n}\ln\frac{x^n}{x^n+1}+C$$</p>
<p>Answer: <strong>(A)</strong></p>
Correct Answer: A