Mathematics

$$\displaystyle\int { \dfrac { \left( x+3 \right) { e }^{ x } }{ { \left( x+4 \right)  }^{ 2 } }  } dx$$ is equal to


ANSWER

$$\dfrac { { e }^{ x } }{ x+4 } +C$$


SOLUTION
$$I=\displaystyle\int { \dfrac { (x+3){ e }^{ x } }{ { (x+4) }^{ 2 } } dx }$$

$$=\displaystyle\int { \dfrac { (x+4-1){ e }^{ x } }{ { (x+4) }^{ 2 } } dx }$$

$$=\displaystyle\int { \dfrac { { e }^{ x } }{ { (x+4) } } dx }$$$$-\displaystyle\int { \dfrac { { e }^{ x } }{ { (x+4) }^{ 2 } } dx }$$

$$=\dfrac{e^x}{x+4}+\displaystyle\int { \dfrac { { e }^{ x } }{ { (x+4)^2 } } dx }$$$$-\displaystyle\int { \dfrac { { e }^{ x } }{ { (x+4) }^{ 2 } } dx }$$ [ Using method of by parts]

$$=\dfrac{e^x}{x+4}+c$$ [ Where $$c$$ is integrating constant].
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Single Correct Medium Published on 17th 09, 2020
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