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need solution for rd sharma maths class 12 chapter Indefinite integrals exercise 18.1 question 6

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Answer: 2\sqrt{x}+c

Hint: Use the formula of e^{\log x}=x

Given: \int \frac{e^{\log \sqrt{x}}}{x}dx

Solution: \int \frac{e^{\log \sqrt{x}}}{x}dx

                =\int \frac{\sqrt{x}}{x}dx                                            \left ( \because e^{\log x} =x\right )

                =\int \frac{1}{\sqrt{x}}dx=\int x^{\frac{-1}{2}}dx

                =\frac{x^{\frac{-1}{2}+1}}{\frac{-1}{2}+1}+c=\frac{x^{\frac{1}{2}}}{\frac{1}{2}}+c                    \left ( \because x^{n}=\frac{x^{n+1}}{n+1} \right )


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