Mathematics

Evaluate: $$\displaystyle \int\dfrac{(x-4)e^x}{(x-2)^3}dx$$.


SOLUTION
Given the integral
$$\int { \dfrac { (x-4){ e }^{ x } }{ { (x-2) }^{ 3 } }  } dx\\ =\int { \dfrac { (x-2-2){ e }^{ (x-2) }{ e }^{ 2 } }{ { (x-2) }^{ 3 } }  } dx\\ ={ e }^{ 2 }\int { \dfrac { (x-2-2){ e }^{ (x-2) } }{ { (x-2) }^{ 3 } }  } dx$$
Let us assume,
$$u=x-2\\ \Rightarrow \dfrac { du }{ dx } =1\\ \Rightarrow du=dx$$
Substituting these values in the integral we get,
$${ e }^{ 2 }\int { \dfrac { (x-2-2){ e }^{ (x-2) } }{ { (x-2) }^{ 3 } }  } dx\\ ={ e }^{ 2 }\int { \dfrac { (u-2){ e }^{ u } }{ { u }^{ 3 } }  } du$$
For $$\int { \dfrac { (u-2){ e }^{ u } }{ { u }^{ 3 } }  } du$$,
$$\int { \dfrac { (u-2){ e }^{ u } }{ { u }^{ 3 } }  } du\\ =\int { \dfrac { u{ e }^{ u } }{ { u }^{ 3 } }  } du-2\int { \dfrac { { e }^{ u } }{ { u }^{ 3 } }  } du\\ =\int { \dfrac { { e }^{ u } }{ { u }^{ 2 } }  } du-2\int { \dfrac { { e }^{ u } }{ { u }^{ 3 } }  } du$$
Using integration by parts for $$\int { \dfrac { { e }^{ u } }{ { u }^{ 3 } }  } du$$
$$=\dfrac { 2{ e }^{ u } }{ 2{ u }^{ 2 } } +2\int { -\dfrac { { e }^{ u } }{ 2{ u }^{ 2 } }  } du+\int { \dfrac { { e }^{ u } }{ { u }^{ 2 } }  } du\\ =\dfrac { { e }^{ u } }{ { u }^{ 2 } } -\int { \dfrac { { e }^{ u } }{ { u }^{ 2 } }  } du+\int { \dfrac { { e }^{ u } }{ { u }^{ 2 } }  } du\\ =\dfrac { { e }^{ u } }{ { u }^{ 2 } } $$
So, 
$${ e }^{ 2 }\int { \dfrac { (u-2){ e }^{ u } }{ { u }^{ 3 } }  } du\\ ={ e }^{ 2 }\dfrac { { e }^{ u } }{ { u }^{ 2 } } \\ =\dfrac { { e }^{ u+2 } }{ { u }^{ 2 } } \quad \quad \left[ \because u=x-2 \right] \\ =\dfrac { { e }^{ x } }{ { (x-2) }^{ 2 } } \\ \therefore \int { \dfrac { (x-4){ e }^{ x } }{ { (x-2) }^{ 3 } }  } dx=\dfrac { { e }^{ x } }{ { (x-2) }^{ 2 } } +C.$$
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Subjective Medium Published on 17th 09, 2020
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