∫x−3(x−1)3⋅exdx is equal to

Let I=∫x−3(x−1)3exdx=∫x−1−2(x−1)3exdx=∫exx−1(x−1)3−2(x−1)3dx=∫ex1(x−1)2−2(x−1)3dxLet f(x)=1(x−1)2⇒f′(x)=−2(x−1)3 ∴ I=ex(x−1)2+C ∵∫exf(x)+f′(x)dx=exf(x)

∫x−3(x−1)3⋅exdx is equal to

Let I=∫x−3(x−1)3exdx=∫x−1−2(x−1)3exdx=∫exx−1(x−1)3−2(x−1)3dx=∫ex1(x−1)2−2(x−1)3dxLet f(x)=1(x−1)2⇒f′(x)=−2(x−1)3 ∴ I=ex(x−1)2+C ∵∫exf(x)+f′(x)dx=exf(x)

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$\int \frac{x - 3}{(x - 1)^{3}} \cdot e^{x} dx$ is equal to

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$\frac{- e^{x}}{(x - 1)^{2}} + C$

$\frac{e^{x}}{(x - 1)} + c$

$\frac{2 e^{x}}{(1 - x)^{2}} + C$

$\frac{e^{x}}{(x - 1)^{2}} + C$

answer is D.

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Detailed Solution

Let $\begin{aligned} I = \int \frac{x - 3}{(x - 1)^{3}} e^{x} dx = \int \frac{x - 1 - 2}{(x - 1)^{3}} e^{x} dx \\ = \int e^{x} [\frac{x - 1}{(x - 1)^{3}} - \frac{2}{(x - 1)^{3}}] dx \end{aligned}$