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Please solve RD Sharma class 12 chapter Indefinite Integrals exercise 18.9 question 61 maths textbook solution

Answers (1)


Answer:2 \tan (\sqrt{x})+c

Hint: Use substitution method to solve this integral

Given: \int \frac{\sec ^{2} \sqrt{x}}{\sqrt{x}} d x


        \text { Let } I=\int \frac{\sec ^{2} \sqrt{x}}{\sqrt{x}} d x

        \begin{aligned} &\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, } I=\int \frac{\sec ^{2} t}{\sqrt{x}} \cdot 2 \sqrt{x}\; d t=2 \int \sec ^{2} t \; d t \end{aligned}

                        =2 \tan t+c \quad\left[\because \int \sec ^{2} x d x=\tan x+c\right]

                        =2 \tan \sqrt{x}+c \quad[\because t=\sqrt{x}]

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