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Please solve RD Sharma class 12 chapter Indefinite Integrals exercise 18.26 question 17 maths textbook solution

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

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

Given:

\int \frac{e^{x}}{x}\left\{x(\log x)^{2}+2 \log x\right\} d x

Solution:

        I=\int \frac{e^{x}}{x}\left\{x(\log x)^{2}+2 \log x\right\} d x

\int e^{x}\left\{(\log x)^{2}+2 \log x \cdot \frac{1}{x}\right\} d x

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

\text { Let } f(x)=(\log x)^{2}, f^{\prime}(x)=2 \log x \cdot \frac{1}{x}

        \begin{aligned} &f(x)+f^{\prime}(x)=(\log x)^{2}+2 \log x \cdot \frac{1}{x} \\ &I=\int e^{x}\left\{f(x)+f^{\prime}(x)\right\} d x \quad \text { if } f(x)=(\log x)^{2} \end{aligned}

            \begin{aligned} &=e^{x} f(x)+c \\ &=e^{x}(\log x)^{2}+c \end{aligned}

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

 

 

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