Indefinite Integration
Integration by Substitution
IIT-JEE 1985
Grade 12
Question:
$\int \frac{dx}{1 + \sqrt{x}}$ equals:
(1) $2[\sqrt{x} - \ln(1 + \sqrt{x})] + C$
(2) $2[\sqrt{x} + \ln(1 + \sqrt{x})] + C$
(3) $\sqrt{x} - \ln(1 + \sqrt{x}) + C$
(4) $2 \ln(1 + \sqrt{x}) + C$
Step-by-Step Solution
Key Concept: Substitute $t = \sqrt{x}$ (so $x = t^2$, $dx = 2t \, dt$) to rationalise the denominator and obtain a simple rational integrand in $t$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)