Find:
         \int \frac{x-5}{\left ( x-3 \right )^{3}}e^{x}dx

 

 

 

 
 
 
 
 

Answers (1)

\int \frac{x-5}{\left ( x-3 \right )^{3}}e^{x}dx
Let I= \int \frac{x-5}{\left ( x-3 \right )^{3}}e^{x}dx
         I= \int\frac{\left ( x-3 \right )-2}{\left ( x-3 \right )^{3}}e^{x}dx\Rightarrow \int \left [ \frac{\left ( x-3 \right )}{\left ( x-3 \right )^{3}}\: \: \frac{-2}{\left ( x-3 \right )^{3}} \right ]e^{x}dx
           \Rightarrow \int \left [ \frac{1}{\left ( x-3 \right )^{2}}\: \: -\frac{2}{\left ( x-3 \right )^{3}} \right ]e^{x}dx-\left ( i \right )
Now,\frac{d}{dx}\left ( x-3 \right )^{-2}= -2\left ( x-3 \right )^{3}= \frac{-2}{\left ( x-3 \right )^{3}}
and we know that
\int \left [ f\left ( x \right ) +{f}'\left ( x \right )\right ]e^{x}dx= e^{x}\left ( f\left ( x \right ) \right )+c
here
 f\left ( x \right )=\frac{1}{\left ( x-3 \right )^{2}} \: \: and\: \: {f}'\left ( x \right )= \frac{-2}{\left ( x-3 \right )^{3}}
Then \int \frac{x-5}{\left ( x-3 \right )^{3}}e^{x}dx= e^{x}\times \frac{1}{\left ( x-3 \right )^{2}}+c= \frac{e^{x}}{\left ( x-3 \right )^{2}}+c

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