Question

\[\int\frac{x}{\sqrt{x + a} - \sqrt{x + b}}dx\]

Answer 1

\[\int\frac{x}{\sqrt{x + a} - \sqrt{x + b}}dx\]
\[ = \int\frac{x}{\sqrt{x + a} - \sqrt{x + b}} \times \frac{\sqrt{x + a} + \sqrt{x + b}}{\sqrt{x + a} + \sqrt{x + b}}dx\]
\[ = \int\frac{x\left( \sqrt{x + a} + \sqrt{x + b} \right)}{\left( \sqrt{x + a} \right)^2 - \left( \sqrt{x + b} \right)^2}dx\]
\[ = ∫  \frac{x\left( \sqrt{x + a} + \sqrt{x + b} \right)}{x + a - x - b}dx\]
\[ = \frac{1}{a - b}\  ∫ x\left( \sqrt{x + a} + \sqrt{x + b} \right) dx\]
\[ = \frac{1}{a - b}\left[  ∫ x \left( \sqrt{x + a} \right) dx + \ ∫x\left( \sqrt{x + b} \right) dx \right]\]
\[ = \frac{1}{a - b}\left[ ∫ \left( x + a - a \right)\left( \sqrt{x + a} \right) dx + \int\left( x + b - b \right)\left( \sqrt{x + b} \right) dx \right]\]
\[ = \frac{1}{a - b}\left[ \int\left( x + a \right)\left( \sqrt{x + a} \right) dx - a\int\left( \sqrt{x + a} \right) dx + \int\left( x + b \right)\left( \sqrt{x + b} \right) dx - b\int\left( \sqrt{x + b} \right) dx \right]\]
\[ = \frac{1}{a - b}\left[ \int \left( x + a \right)^\frac{3}{2} dx - a\int \left( x + a \right)^\frac{1}{2} dx + \int \left( x + b \right)^\frac{3}{2} dx - b\int \left( x + b \right)^\frac{1}{2} dx \right]\]
\[ = \frac{1}{a - b}\left[ \frac{\left( x + a \right)^\frac{5}{2}}{\frac{5}{2}} - a\frac{\left( x + a \right)^\frac{3}{2}}{\frac{3}{2}} + \frac{\left( x + b \right)^\frac{5}{2}}{\frac{5}{2}} - b\frac{\left( x + b \right)^\frac{3}{2}}{\frac{3}{2}} \right] + \text{c             where, c is an arbitrary constant}\]
\[ = \frac{1}{a - b}\left[ \frac{2}{5} \left( x + a \right)^\frac{5}{2} - \frac{2a}{3} \left( x + a \right)^\frac{3}{2} + \frac{2}{5} \left( x + b \right)^\frac{5}{2} - \frac{2b}{3} \left( x + b \right)^\frac{3}{2} \right] + \text{c             where, c is an arbitrary constant}\]
\[Hence, \int\frac{x}{\sqrt{x + a} - \sqrt{x + b}}dx = \frac{1}{a - b}\left[ \frac{2}{5} \left( x + a \right)^\frac{5}{2} - \frac{2a}{3} \left( x + a \right)^\frac{3}{2} + \frac{2}{5} \left( x + b \right)^\frac{5}{2} - \frac{2b}{3} \left( x + b \right)^\frac{3}{2} \right] + \text{c           where, c is an arbitrary constant}\]