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need solution for RD Sharma maths class 12 chapter Indefinite Integrals exercise 18.9 question 3

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Answer: \frac{2}{3}(1+\sqrt{x})^{3}+c

Hint: Use substitution method to solve this integral.

Given: \int \frac{(1+\sqrt{x})^{2}}{\sqrt{x}} d x


        Let I=\int \frac{(1+\sqrt{x})^{2}}{\sqrt{x}} d x

        Put 1+\sqrt{x}=t \Rightarrow \frac{1}{2 \sqrt{x}} d x=d t

        \begin{aligned} &\Rightarrow d x=2 \sqrt{x} d t \text { then } \\ &I=\int \frac{t^{2}}{\sqrt{x}} 2 \sqrt{x} d t=\int 2 t^{2} d t=2 \int t^{2} d t \end{aligned}

        \begin{aligned} &=2 \frac{t^{2+1}}{2+1}+c=2 \frac{t^{3}}{3}+c \quad\left[\because \int x^{n} d x=\frac{x^{n+1}}{n+1}+c\right] \\ &=\frac{2}{3}(1+\sqrt{x})^{3}+c \quad[\because t=1+\sqrt{x}] \end{aligned}

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