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need solution for RD Sharma maths class 12 chapter Indefinite Integrals exercise 18.26 question 23

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Answer:
The correct answer is  e^{x} \frac{1}{(x-2)^{2}}+c
Hint:

\int e^{x}\left(f(x)+f^{\prime}(x)\right) d x=e^{x} f(x)+c

Given:

\int e^{x} \frac{(x-4)}{(x-2)^{3}} d x

Solution:

        \begin{aligned} &I=\int e^{x} \frac{(x-4)}{(x-2)^{3}} d x \\ &=\int e^{x} \frac{(x-2-2)}{(x-2)^{3}} d x \\ &=\int e^{x}\left[\frac{(x-2)}{(x-2)^{3}}-\frac{2}{(x-2)^{3}}\right] d x \end{aligned}

        =\int e^{x}\left[\frac{1}{(x-2)^{2}}+\frac{-2}{(x-2)^{3}}\right] d x

Now, consider,

        f(x)=\frac{1}{(x-2)^{2}}

Then  f^{\prime}(x)=\frac{-2}{(x-2)^{3}}

Thus the above integral is of the form

        e^{x}\left(f(x)+f^{\prime}(x)\right)

As,

        \int e^{x}\left(f(x)+f^{\prime}(x)\right) d x=e^{x} f(x)+c

        I=e^{x} \frac{1}{(x-2)^{2}}+c

So, the correct answer is  e^{x} \frac{1}{(x-2)^{2}}+c

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