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need solution for RD Sharma maths class 12 chapter Indefinite Integrals exercise 18.9 question 51

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Answer: 2 \sin \sqrt{x}+c

Hint: Use substitution method to solve this integral.

Given:   \int \frac{\cos \sqrt{x}}{\sqrt{x}} d x


        \begin{aligned} &\text { Let } I=\int \frac{\cos \sqrt{x}}{\sqrt{x}} d x \\ &\text { Put } \sqrt{x}=t \Rightarrow \frac{1}{2 \sqrt{x}} d x=d t \\ &\Rightarrow d x=2 \sqrt{x}\; d t \text { then } \end{aligned}

        \begin{aligned} I &=\int \frac{\cos t}{\sqrt{x}} \cdot 2 \sqrt{x} \; d t \Rightarrow 2 \int \cos t \; d t \\ &=2 \sin t+\mathrm{c} \quad\left[\because \int \cos x\; d x=\sin x+c\right] \\ &=2 \sin \sqrt{x}+c \quad[\because t=\sqrt{x}] \end{aligned}

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