Q115 / 180 · KCET 2008 · Mathematics
If $\int f(x) d x=g(x)$, then $\int f(x) g(x) d x$ is equal to
- A$\frac{1}{2} \mathrm{f}^{2}(\mathrm{x})$
- Correct answerB$\frac{1}{2} g^{2}(x)$
- C$\frac{1}{2}\left[g^{\prime}(x)\right]^{2}$
- D$f^{\prime}(x) g(x)$
Solution
Given, $\int f(x) d x=g(x)$
$-\int\left[g^{\prime}(x) \int f(x) d x\right] d x$
$$
=g(x) g(x)-\int g^{\prime}(x) g(x) d x
$$
$$
\begin{aligned}
&=[g(x)]^{2}-\frac{[g(x)]^{2}}{2} \\
&=\frac{g^{2}(x)}{2}
\end{aligned}
$$
Hence, the correct answer is option: 'B'.