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Q20  Integrate the functions \frac{( x-3)e ^x }{( x-1)^3}

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\frac{( x-3)e ^x }{( x-1)^3}
It is known that \int e^x[f(x)+f'(x)]=e^x[f(x)]+C

So, By adjusting the given equation, we get
\int\frac{( x-3)e ^x }{( x-1)^3} =\int e^x(\frac{x-1-2}{(x-1)^3}) =\int e^x({\frac{1}{(x-1)^2}-\frac{2}{(x-1)^3})}dx

to let 
f(x)=\frac{1}{(x-1)^2}\Rightarrow f'(x)=-\frac{2}{(x-1)^3}
Therefore the required solution of the givenI=\frac{e^x}{(x-1)^2}+C integral is 


 

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manish

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