Indefinite Integration
Integration by Substitution
Grade 12
Question:
<p>\(\int \frac{x+1}{x\sqrt{x+1}} dx\) is equal to</p>
<p>(A) \(\ln|x - \sqrt{x+1}| - \tan^{-1}x + C\)</p>
<p>(B) \(\ln|x + \sqrt{x+1}| - \tan^{-1}x + C\)</p>
<p>(C) \(\ln|x - \sqrt{x+1}| + \tan^{-1}x + C\)</p>
<p>(D) None of these</p>
Step-by-Step Solution
Key Concept: Use substitution $u = \sqrt{x+1}$ to rationalize and simplify the integrand.
<p>Let $u = \sqrt{x+1}$, then $u^2 = x+1$ and $2u\,du = dx$. The integral becomes manageable and evaluates to the logarithmic and inverse tangent combination.</p>
Correct Answer: A