Integrals — Class 12 Maths Solution

.: Let $I = \int {\cfrac{1}{{\sqrt {\left( {x - a} \right)\left( {x - b} \right)} }}dx} = \int {\cfrac{{dx}}{{\sqrt {{x^2} - \left( {a + b} \right)x + ab} }}}$

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

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

$= \log \left| {\left( {x - \cfrac{{a + b}}{2}} \right) + \sqrt {{{\left( {x - \cfrac{{a + b}}{2}} \right)}^2} - {{\left( {\cfrac{{a - b}}{2}} \right)}^2}} } \right| + C$

$= \log \left| {\left( {x - \cfrac{{a + b}}{2}} \right) + \sqrt {\left( {x - a} \right)\left( {x - b} \right)} } \right| + C$