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Evaluate : $ \int \large\frac{(x-4)e^x}{(x-2)^3}$$dx $

1 Answer

Toolbox:
  • $\int e^x[f(x)+f'(x)dx]=e^xf(x)+c$
Step 1:
Let $I=\int \large\frac{(x-4)e^x dx}{(x-2)^3}$
This can be written as
$\qquad= \int \large\frac{(x-2)-2}{(x-2)^3}$$dx$
$\qquad=\int\bigg[\large\frac{1}{(x-2)^2}-\frac{2}{(x-2)^3}\bigg]$$dx$
Step 2:
This is of the form
$\qquad =\int e^x[f(x)+f'(x)]dx$
$\qquad=e^x(f(x))$
Here $f(x)=\large\frac{1}{(x-2)^2}$
$f'(x)=\large\frac{-2}{(x-2)^2}$
$\therefore I=\large\frac{e^x}{(x-2)^2}$
answered Sep 23, 2013 by sreemathi.v
 
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