$\int \sin \sqrt{x} \mathrm{~d} x=\ldots+C$ (where $C$ is a constant of integration.)

$\int \sin \sqrt{x} \mathrm{~d} x=\ldots+C$ (where $C$ is a constant of integration.)
  1. $2(-\sqrt{x} \cos \sqrt{x}+\sin \sqrt{x})$
  2. $2(-\cos \sqrt{x}+\sin \sqrt{x})$
  3. $2(\cos \sqrt{x}+\sqrt{x} \sin \sqrt{x})$
  4. $2(\sqrt{x} \cos \sqrt{x}+\sin \sqrt{x})$

Solution

$\begin{aligned} & \int \sin \sqrt{x} \mathrm{~d} x \text { let } \sqrt{x}=t \\ & \Rightarrow \mathrm{d} x=2 t \mathrm{~d} t \\ & =2 \int t \sin t \mathrm{~d} t \\ & =2\left\{t \int \sin t \mathrm{~d} t-\int\left(\frac{\mathrm{d} t}{\mathrm{~d} t} \int \sin t \mathrm{~d} t\right) \mathrm{d} t\right\} \text { [integrating by parts] } \\ & =2\left\{t(-\cos t)+\int \cos t \mathrm{~d} t\right\} \\ & =2\{-t \cos t+\sin t\}+C \\ & =2\{-\sqrt{x} \cos \sqrt{x}+\sin \sqrt{x}\}+C\end{aligned}$

Asked in: MHT CET 2022 (08 Aug Shift 2)

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