Mathematics

Solve $$\displaystyle\int {\frac{{{e^x}\left( {{x^2} + 5x + 7} \right)}}{{{{\left( {x + 3} \right)}^2}}}} dx$$


SOLUTION
$$\displaystyle\int {\dfrac{{{e^x}\left( {{x^2} + 5x + 7} \right)}}{{{{\left( {x + 3} \right)}^2}}}} dx$$

let
$${u^2} = {\left( {x + 3} \right)^2}\\$$

$$\implies {x^2} + 5x + 7 = {u^2} - u + 1\\$$

$$\displaystyle \int {\dfrac{{{e^{u - 3}}\left( {{u^2} - u + 1} \right)}}{{{{\left( u \right)}^2}}}} du\\$$

$$\displaystyle={e^{-3}}\int {{e^u}\left( {\dfrac{1}{{{u^2}}} - \dfrac{1}{u} + 1} \right)du\,} \, $$


$$\displaystyle= {e^{ - 3}}\int {{e^u}du} \, + {e^{ - 3}}\int {\left( {\dfrac{1}{u} - \dfrac{1}{{{u^2}}}} \right){e^u}du} \,\\ $$

$$\displaystyle= {e^{ - 3}}{e^u} + \dfrac{{{e^{ - 3}}}}{u} $$


$$= {e^{u - 3}} + \dfrac{{{e^{ - 3}}}}{{u }} $$


Putting the value of u, we get

$$= {e^x} + \dfrac{{{e^{ - 3}}}}{{x + 3}} + c$$
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Subjective Medium Published on 17th 09, 2020
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