Q120 / 180 · COMEDK 2019 · Mathematics

If $\int f(x) d x=g(x)$, then $\int f(x) g(x) d x=$

  • A
    $\frac{1}{2} f^{2}(x)$
  • Correct answer
    B
    $\frac{1}{2} g^{2}(x)$
  • C
    $\frac{1}{2}\left[g^{\prime}(x)\right]^{2}$
  • D
    $f^{\prime}(x) g(x)$

Solution

We have, $\int f(x) d x=g(x)$ $\Rightarrow \quad f(x)=g^{\prime}(x)$ Now, let $I=\int f(x) \cdot g(x) d x=\int g^{\prime}(x) g(x) d x$ Put $g(x)=t$ $\Rightarrow \quad g^{\prime}(x) d x=d t$ $\therefore \quad I=\int t d t=\frac{t^{2}}{2}+C=\frac{1}{2}(g(x))^{2}+C$

Hence, the correct answer is option: 'B'.

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