JEE MainMathematicsEasyIndefinite integration

$\int\frac{dx}{\sqrt{(1 + x)} + \sqrt{x}} =$

Show answer and solution

Rationalising the denominator, we have

\int\frac{dx}{\sqrt{(1 + x)} + \sqrt{x}} = \int\frac{\{\sqrt{(1 + x)} - \sqrt{x}\}}{\{\sqrt{(1 + x)} + \sqrt{x}\}\{\sqrt{(1 + x)} - \sqrt{x}\}}dx

= \int\frac{\{\sqrt{(1 + x)} - \sqrt{x}\}}{(1 + x) - x}dx = \int\lbrack\sqrt{(1 + x)} - \sqrt{x}\rbrack dx

= \frac{2}{3}(1 + x)^{\frac{3}{2}} - \frac{2}{3}x^{\frac{3}{2}} + c

Source: Grass-G / M-12 Mathematics Paper 2 Nov.docx

Need a video for this idea?

Scan any doubt from your notes and get a whiteboard explanation.