KCET · Maths · Indefinite Integration
If \(\int f(x) d x=g(x)\), then \(\int f(x) g(x) d x\) is equal to
- A \(\frac{1}{2} \mathrm{f}^{2}(\mathrm{x})\)
- B \(\frac{1}{2} g^{2}(x)\)
- C \(\frac{1}{2}\left[g^{\prime}(x)\right]^{2}\)
- D \(f^{\prime}(x) g(x)\)
Answer & Solution
Correct Answer
(B) \(\frac{1}{2} g^{2}(x)\)
Step-by-step Solution
Detailed explanation
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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