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Please solve RD Sharma class 12 chapter Indefinite Integrals exercise 18.9 question 45 maths textbook solution

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Answer: 2 \sin \left(e^{\sqrt{x}}\right)+c

Hint: Use substitution method to solve this integral.

Given:   \int \frac{e^{\sqrt{x}} \cos \left(e^{\sqrt{x}}\right)}{\sqrt{x}} d x


        \begin{aligned} &\text { Let } I=\int \frac{e^{\sqrt{x}} \cos \left(e^{\sqrt{x}}\right)}{\sqrt{x}} d x \\ &\text { Put } e^{\sqrt{x}}=t \Rightarrow e^{\sqrt{x}} \frac{1}{2 \sqrt{x}} d x=d t \\ &\Rightarrow d x=\frac{2 \sqrt{x}}{e^{\sqrt{x}}} \mathrm{dt} \text { then } \end{aligned}

        \begin{aligned} &I=\int \frac{e^{\sqrt{x}} \cos t}{\sqrt{x}} \frac{2 \sqrt{x}}{e^{\sqrt{x}}} d t \\ &=2 \int \cos t \; d t \end{aligned}

         =2 \sin t+c \quad\left[\because \int \cos x\; d x=\sin x+c\right]

         =2 \sin \left(e^{\sqrt{x}}\right)+c \quad\left[\because t=e^{\sqrt{x}}\right]

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