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need solution for RD Sharma maths class 12 chapter Indefinite Integrals exercise 18.26 question 15

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Answer:
The correct answer is \log x\; e^{x}+c
Given:

\int e^{x}\left(\log x+\frac{1}{x}\right) d x

Solution:

        I=\int e^{x}\left(\log x+\frac{1}{x}\right) d x

            =\int e^{x} \log x \; d x+\int e^{x} \cdot \frac{1}{x} d x

On integration by parts,

            \begin{aligned} &\log x \; e^{x}-\int \frac{1}{x} \cdot e^{x} d x+\int e^{x} \cdot \frac{1}{x} d x \\ &=\log x \; e^{x}+c \end{aligned}

So, the correct answer is \log x\; e^{x}+c

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