Integrals — Class 12 Maths Solution

: $\int\limits_0^1 {\cfrac{{2x + 3}}{{5{x^2} + 1}}dx} = \int\limits_0^1 {\left( {\cfrac{{2x}}{{5{x^2} + 1}} + \cfrac{3}{{5{x^2} + 1}}} \right)dx}$

$= \cfrac{1}{5}\int\limits_0^1 {\cfrac{{10x}}{{5{x^2} + 1}}} dx + \cfrac{3}{5}\int\limits_0^1 {\cfrac{{dx}}{{{x^2} + {{\left( {\cfrac{1}{{\sqrt 5 }}} \right)}^2}}}}$

$= \cfrac{1}{5}\left[ {\log \left( {5{x^2} + 1} \right)} \right]_0^1 + \cfrac{3}{5} \times \cfrac{1}{{\cfrac{1}{{\sqrt 5 }}}}\left[ {{{\tan }^{ - 1}}\left( {\cfrac{x}{{\cfrac{1}{{\sqrt 5 }}}}} \right)} \right]_0^1$

$= \cfrac{1}{5}\left( {\log 6 - \log 1} \right) + \cfrac{3}{{\sqrt 5 }}\left( {{{\tan }^{ - 1}}\sqrt 5 - 0} \right) = \cfrac{1}{5}\log 6 + \cfrac{3}{{\sqrt 5 }}{\tan ^{ - 1}}\sqrt 5$

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