Evaluate : ∫(x-4)e^x/(x-2)^3

Question

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

Answer 1

I = \(\int\begin{bmatrix} \frac{(x-2)-2}{(x-2)^3} \\[0.3em] \end{bmatrix}e^xdx\)

\(= \int\frac{e^x}{(x-2)^2}dx-\)\(2\int\frac{e^x}{(x-2)^3}dx\)

\(=\frac{e^x}{(x-2)^2}+\)\(2\int\frac{e^x\,dx}{(x-2)^3}-\)\(2\int\frac{e^x\,dx}{(x-2)^3}\)

\(=\frac{e^x}{(x-2)^2}+\) C