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  • Provide solution for rd sharma maths class 12 chapter Indefinite Integrals exercise 18.2 question 14

Provide solution for rd sharma maths class 12 chapter Indefinite Integrals exercise 18.2 question 14

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2 \sqrt{x}+2 x+\frac{2}{3} x^{\frac{3}{2}}+c

Hint:To solve this equation \left ( 1+\sqrt{x} \right ) will be differentiate first

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

Solution: We have

\begin{aligned} &\int \frac{(1+\sqrt{x})^{2}}{\sqrt{x}}^{\frac{1}{x^{\frac{1}{2}}}} d x=\int \frac{1+x+2 \sqrt{x}}{x^{\frac{1}{2}}}+\left[\because(a+b)^{2}=a^{2}+2 a b+b^{2}\right] \\ &=\int \left ( \frac{1}{x^{\frac{1}{2}}}+\frac{x}{x^{\frac{1}{2}}}+\frac{2\sqrt{x}}{\sqrt{x}} \right )dx\\ &=\int x^{\frac{-1}{2}} d x+\int x^{\frac{1}{2}} d x+2 \int d x \\ &=\frac{x^{\frac{-1}{2}+1}}{\frac{-1}{2}+1}+\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}+2 x+c \\ &=2 \sqrt{x}+\frac{2}{3} x^{\frac{3}{2}}+2 x+c \end{aligned}

 

 

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