Integrals — Class 12 Maths Solution

Let $I = \int {\cfrac{1}{{\sqrt {x + a} + \sqrt {x + b} }}} dx$

$= \int {\cfrac{1}{{\sqrt {x + a} + \sqrt {x + b} }} \times \cfrac{{\sqrt {x + a} - \sqrt {x + b} }}{{\sqrt {x + a} - \sqrt {x + b} }}} dx$

$= \int {\cfrac{{\sqrt {x + a} - \sqrt {x + b} }}{{{{\left( {\sqrt {x + a} } \right)}^2} - {{\left( {\sqrt {x + b} } \right)}^2}}}dx\cfrac{1}{{a - b}}\int {\left[ {{{\left( {x + a} \right)}^{1/2}} - {{\left( {x + b} \right)}^{1/2}}} \right]} } dx$

$= \cfrac{1}{{a - b}}\left[ {\cfrac{{{{\left( {x + a} \right)}^{3/2}}}}{{\cfrac{3}{2}}} - \cfrac{{{{\left( {x + b} \right)}^{3/2}}}}{{\cfrac{3}{2}}}} \right] + C$

$= \cfrac{2}{{3\left( {a - b} \right)}}\left[ {{{\left( {x + a} \right)}^{3/2}} - {{\left( {x + b} \right)}^{3/2}}} \right] + C$