Mathematics - Application of Derivative Question with Solution | TestHub

$$\begin{array}{l}\int f(x)dx=f(x)\\ \Rightarrow \frac{d}{dx}f(x)=f(x)\\ \left[\because f(x)=\int \frac{d}{dx}f(x)dx\right]\end{array}$$ Now, $\int \{f(x){\}}^{2}dx=\int f(x)\cdot f(x)dx$ $=f(x)\int f(x)dx-\int \left[\frac{d}{dx}f(x)\int f(x)dx\right]dx$ (integrating by parts) $$\begin{array}{l}=f(x)f(x)-\int f(x)f(x)dx\\ \Rightarrow 2\int \{f(x){\}}^{2}dx=\{f(x){\}}^2\\ \Rightarrow \int \{f(x){\}}^{2}dx=\frac{1}{2}\{f(x){\}}^2\end{array}$$