If ∫f(x)dx=f(x)+C, then ∫{f(x)}2dx is

# If  is

1. A

$\left\{\mathrm{f}\left(\mathrm{x}\right){\right\}}^{2}+\mathrm{C}$

2. B

$\frac{1}{3}\left\{\mathrm{f}\left(\mathrm{x}\right){\right\}}^{3}+\mathrm{C}$

3. C

$\mathrm{f}\left(\mathrm{x}\right)+\mathrm{C}$

4. D

$\frac{1}{2}\left\{\mathrm{f}\left(\mathrm{x}\right){\right\}}^{2}+\mathrm{C}$

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### Solution:

Given,

$\begin{array}{l}\int \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\mathrm{f}\left(\mathrm{x}\right)+\mathrm{C}\\ \mathrm{f}\left(\mathrm{x}\right)={\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)\\ \therefore \int \left\{\mathrm{f}\left(\mathrm{x}{\right)}^{2}\right\}\mathrm{dx}=\int \mathrm{f}\left(\mathrm{x}\right)\cdot {\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)\mathrm{dx}=\frac{1}{2}\left\{\mathrm{f}\left(\mathrm{x}\right){\right\}}^{2}+\mathrm{C}\end{array}$

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