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  • Explain solution rd sharma class 12 chapter Indefinite Integrals exercise 18.2 question 45

Explain solution rd sharma class 12 chapter Indefinite Integrals exercise 18.2 question 45

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\frac{x^{2}}{2}+\frac{1}{x}+c\\ Hint: Using\;\; \int x^{n} d x\\ Given: If \;\; f^{\prime}(x)=x-\frac{1}{x^{2}} \;\;\; and \;\;\; f(1)=\frac{1}{2}, \;\;\; find \;\;\;f(x)\\ Solution: f^{\prime}(x)=x-\frac{1}{x^{2}}\\

Integrating both side

f(x)=\int\left(x-\frac{1}{x^{2}}\right) d x

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

\begin{aligned} &\text { We have } f(1)=\frac{1}{2}\\ &\Rightarrow \frac{(1)^{2}}{2}+\frac{1}{1}+c=\frac{1}{2}\\ &\Rightarrow \frac{1}{2}+\frac{1}{1}+c=\frac{1}{2}\\ &\Rightarrow c=-1\\ &\text { Put in (1) }\\ &\therefore f(x)=\frac{x^{2}}{2}+\frac{1}{x}-1 \end{aligned}

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