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If $\int f(x) d x=f(x)$, then $\int\{f(x)\}^{2} d x$ is equal to
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Verified Answer
The correct answer is:
$\frac{1}{2}\{\mathrm{f}(\mathrm{x})\}^{2}$
$$
\begin{array}{l}
\int f(x) d x=f(x) \\
\Rightarrow \frac{d}{d x} f(x)=f(x) \\
\quad\left[\because f(x)=\int \frac{d}{d x} f(x) d x\right]
\end{array}
$$
Now, $\int\{f(x)\}^{2} d x=\int f(x) \cdot f(x) d x$ $=f(x) \int f(x) d x-\int\left[\frac{d}{d x} f(x) \int f(x) d x\right] d x$ (integrating by parts)
$$
\begin{array}{l}
=f(x) f(x)-\int f(x) f(x) d x \\
\Rightarrow 2 \int\{f(x)\}^{2} d x=\{f(x)\}^{2} \\
\Rightarrow \int\{f(x)\}^{2} d x=\frac{1}{2}\{f(x)\}^{2}
\end{array}
$$
\begin{array}{l}
\int f(x) d x=f(x) \\
\Rightarrow \frac{d}{d x} f(x)=f(x) \\
\quad\left[\because f(x)=\int \frac{d}{d x} f(x) d x\right]
\end{array}
$$
Now, $\int\{f(x)\}^{2} d x=\int f(x) \cdot f(x) d x$ $=f(x) \int f(x) d x-\int\left[\frac{d}{d x} f(x) \int f(x) d x\right] d x$ (integrating by parts)
$$
\begin{array}{l}
=f(x) f(x)-\int f(x) f(x) d x \\
\Rightarrow 2 \int\{f(x)\}^{2} d x=\{f(x)\}^{2} \\
\Rightarrow \int\{f(x)\}^{2} d x=\frac{1}{2}\{f(x)\}^{2}
\end{array}
$$
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