Question

$\int \: \frac{x - 3}{ ({x - 1}^{3} ) } {e}^{x} dx$

Integration
​

Answer 1

$\huge \mathfrak{solution}$

Let

$I = \int \frac{x - 3}{( {x - 1)}^{3} } {e}^{x} dx \\ \\ \rightarrow \int \: \frac{x - 1 - 2}{( {x - 1)}^{3} } {e}^{x} dx \\ \\ \rightarrow \int \: {e}^{x} ( \frac{x - 1}{( {x - 1)}^{3} } - \frac{2}{( {x - 1)}^{3} } )dx \\ \\ \rightarrow \int {e}^{x} ( \frac{1}{( {x - 1)}^{2} } - \frac{2}{(x - 1)^{3} } dx$

Now ,

Let

$f(x) = \frac{1}{ {(x - 1) }^{2} }$

differentiate w.r.t x we get ,

$\implies \: \frac{ - 2}{ {(x - 1)}^{3} }$

As we know that :

$\boxed{ \red{\bold{ \int \: {e}^{x} (f(x) + \frac{d}{dx} (f(x)))dx = {e}^{x} f(x)}}}$

Now compare with this we get ,

$\bold{I = \frac{ {e}^{x} }{ {(x - 1)}^{2} } + c}$

Answer 2

Answer:

$\large\boxed{\sf{ \frac{ {e}^{x} }{ {(x - 1)}^{2} } + c}}$

Step-by-step explanation:

W have to integrate the following expression,

$\displaystyle \int \frac{x - 3}{ {(x - 1)}^{3} } {e}^{x} \: dx$

On Simplifying,

$= \sf{ \displaystyle \int \frac{x - 1 - 2}{ {(x - 1)}^{3} } {e}^{x} \: dx} \\ \\ \sf{ = \displaystyle \int {e}^{x} ( \frac{x - 1}{ {(x - 1)}^{3} } - \frac{2}{ {(x - 1)}^{3} } ) \: dx} \\ \\ \sf{= \displaystyle \int {e}^{x} ( \frac{1}{ {(x - 1)}^{2} } - \frac{2}{ {(x - 1)}^{3} } ) \: dx}$

Now, let's assume that,

$\sf{g(x) = \dfrac{1}{ {(x - 1)}^{2} }}$

Differentiating both sides,

$\sf{= > {g}^{'}(x) = - \dfrac{2}{ {(x - 1)}^{3} }}$

Therefore, We have,

$\sf{= \displaystyle \int {e}^{x} (g(x) + {g}^{'} (x)) \: dx} \\ \\ \sf{ = {e}^{x} g(x) + c} \\ \\ \sf{ = {e}^{x} \times \frac{1}{ {(x - 1)}^{2} } + c} \\ \\ \sf{ = \frac{ {e}^{x} }{ {(x - 1)}^{2} } + c}$

Concept Map :-

• $\sf{\displaystyle \int {e}^{x} (g(x) + {g}^{'} (x)) \: dx = {e}^{x} g(x) + c}$