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Question
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
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Solution
Let `I = ((x - 3) e^x)/(x - 1)^3 dx`
`= int (e^x (x - 1 - 2))/(x - 1)^3 dx`
`= int e^x [1/((x - 1)^2) - 2/((x - 1)^3)] dx`
On substituting `e^x . 1/((x - 1)^2) = t`
`[e^x - 2 (x - 1)^-3 + 1/((x - 1)^2). e^x] dx = dt`
or `e^x [1/((x - 1)^2) - 2/(x - 1)^3] dx = dt`
Hence, `I = int 1. dt = t + C`
`= e^x/((x - 1)^2) + C`
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