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\int { \frac { { \left( x+3 \right) e }^{ x } }{ (x+5)^{ 3 } } dx }

Note by Abhay Raj Singh
2 years, 6 months ago

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

$$= \int \frac { ( x+5 - 2) e^x }{ (x+5)^3 } dx$$

$$= \int \frac { (x+5) e^x }{ (x+5)^3 } dx + \int \frac { - 2 e^x }{ (x+5)^3 } dx$$

$$= \int \frac { e^x }{ (x+5)^2 } dx + \int \frac { - 2 e^x }{ (x+5)^3 } dx$$

Using IBP on the second integral with$$u = e^x$$ and $$dv = \frac{- 2}{(x+5)^3}$$

$$= \int \frac { e^x }{ (x+5)^2 } dx + \frac{e^x}{(x+5)^2} -\int \frac { e^x }{ (x+5)^2 } dx$$

$$= \frac{e^x}{(x+5)^2}$$ · 2 years, 6 months ago