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If $\int f(x) d x=g(x)$, then $\int f(x) g(x) d x$ is equal to
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Verified Answer
The correct answer is:
$\frac{1}{2} g^{2}(x)$
Given, $\int f(x) d x=g(x)$
$-\int\left[g^{\prime}(x) \int f(x) d x\right] d x$
$$
=g(x) g(x)-\int g^{\prime}(x) g(x) d x
$$
$$
\begin{aligned}
&=[g(x)]^{2}-\frac{[g(x)]^{2}}{2} \\
&=\frac{g^{2}(x)}{2}
\end{aligned}
$$
$-\int\left[g^{\prime}(x) \int f(x) d x\right] d x$
$$
=g(x) g(x)-\int g^{\prime}(x) g(x) d x
$$
$$
\begin{aligned}
&=[g(x)]^{2}-\frac{[g(x)]^{2}}{2} \\
&=\frac{g^{2}(x)}{2}
\end{aligned}
$$
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