#### Explain solution rd sharma class 12 chapter Indefinite Integrals exercise 18.2 question 46

\begin{aligned} &\frac{x^{2}}{2}+\frac{13}{2} x-2\\ &\text { Hint: } \mathrm{U} \operatorname{sing} \int x^{n} d x\\ &\text { Given: If } f^{\prime}(x)=x+b \text { and } f(1)=5, \text { find } f(x) \end{aligned}

\begin{aligned} &\text { Solution: } f^{\prime}(x)=x+b\\ &\text { Integrating both sides. }\\ &f(x)=\int(x+b) d x\\ &f(x)=\frac{x^{2}}{2}+b x+c \quad \text { ..... } \end{aligned}

\begin{aligned} &\text { We have } f(1)=5, f(2)=13\\ &\therefore f(x)=\frac{(1)^{2}}{2}+b(1)+c=5\\ &\Rightarrow \frac{1}{2}+b+c=5 \Rightarrow b+c=5-\frac{1}{2} \Rightarrow b+c=\frac{9}{2} \quad \ldots \ldots .(2)\\ &\text { Also } f(2)=13\\ &\Rightarrow \frac{2^{2}}{2}+b(2)+c=13\\ &=2+2 b+c=13 \Rightarrow 2 b+c=11 \end{aligned}\begin{aligned} &\text { Solving (2) and (3) }\\ &b+c=\frac{9}{2}\\ &2 b+c=11\\ &-b \quad=\frac{9}{2}-11\\ &\Rightarrow-b=\frac{-13}{2}\Rightarrow b=\frac{13}{2} \\ &\text { Put in (3) } \\ &2\left(\frac{13}{2}\right)+c=11 \Rightarrow 13+c=11 \Rightarrow c=-2 \end{aligned}

\begin{aligned} &\text { Put the values in (1); We get }\\ &f(x)=\frac{x^{2}}{2}+\frac{13}{2} x-2 \end{aligned}