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  • Explain solution rd sharma class 12 chapter Indefinite Integrals exercise 18.2 question 49

Explain solution rd sharma class 12 chapter Indefinite Integrals exercise 18.2 question 49

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\begin{aligned} & \frac{2}{3} x^{\frac{3}{2}}+2 x^{\frac{1}{2}}+c \\ &\text { Hint: } \mathrm{U} \operatorname{sing} \int x^{n} d x \end{aligned}

Given: Write the primitive or antiderivative of f(x)=\sqrt{x}+\frac{1}{\sqrt{x}}

\begin{aligned} &\text { Solution: } f(x)=\sqrt{x}+\frac{1}{\sqrt{x}}=x^{\frac{1}{2}}+\frac{1}{x^{\frac{1}{2}}} \\ &f(x)=x^{\frac{1}{2}}+x^{-\frac{1}{2}} \end{aligned}

Integrating both sides

\begin{aligned} &\int f(x)=\int\left(x^{\frac{1}{2}}+x^{-\frac{1}{2}}\right) d x \\ &=\int x^{\frac{1}{2}} d x+\int x^{-\frac{1}{2}} d x \\ &=\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}+\frac{x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}+c\\ &=\frac{x^{\frac{3}{2}}}{\frac{3}{2}}+\frac{x^{\frac{1}{2}}}{\frac{1}{2}}+c \\ &=\frac{2}{3} x^{\frac{3}{2}}+2 x^{\frac{1}{2}}+c \end{aligned}

 

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