$\int\left(f(x) \mathrm{g}^{\prime \prime}(x)-\mathrm{f}^{\prime \prime}(x) \mathrm{g}(x)\right) \mathrm{d}…

$\int\left(f(x) \mathrm{g}^{\prime \prime}(x)-\mathrm{f}^{\prime \prime}(x) \mathrm{g}(x)\right) \mathrm{d} x$ is equal to
  1. $\mathrm{f}(x) \mathrm{g}(x)-\mathrm{f}^{\prime}(x) \mathrm{g}^{\prime}(x)$
  2. $\mathrm{f}^{\prime}(x) \mathrm{g}(x)-\mathrm{f}(x) \mathrm{g}^{\prime}(x)$
  3. $\mathrm{f}(x) \mathrm{g}^{\prime}(x)-\mathrm{f}^{\prime}(x) \mathrm{g}(x)$
  4. $\mathrm{f}(x) \mathrm{g}^{\prime}(x)+\mathrm{f}^{\prime}(x) \mathrm{g}(x)$

Solution

$\begin{aligned} & \int\left[\mathrm{f}(x) \mathrm{g}^{\prime \prime}(x)-\mathrm{f}^{\prime \prime}(x) \mathrm{g}(x)\right] \mathrm{d} x \\ & =\mathrm{f}(x) \mathrm{g}^{\prime}(x)-\int \mathrm{f}^{\prime}(x) \mathrm{g}^{\prime}(x) \mathrm{d} x-\mathrm{g}(x) \mathrm{f}^{\prime}(x) \\ & =\mathrm{f}(x) \mathrm{g}^{\prime}(x)-\mathrm{g}(x) \mathrm{f}^{\prime}(x)+\mathrm{c}+\int \mathrm{f}^{\prime}(x) \mathrm{g}^{\prime}(x) \mathrm{d} x\end{aligned}$

Asked in: MHT CET 2024 (02 May Shift 2)

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