A) \[2[\sqrt{x}\sin \sqrt{x}+\cos \sqrt{x}]+c\]
B) \[2[\sqrt{x}\sin \sqrt{x}-\cos \sqrt{x}]+c\]
C) \[2[\cos \sqrt{x}-\sqrt{x}\sin \sqrt{x}]+c\]
D) \[-2[\sqrt{x}\sin \sqrt{x}+\cos \sqrt{x}]+c\]
Correct Answer: A
Solution :
Put \[\sqrt{x}=t\Rightarrow \frac{1}{2\sqrt{x}}\,dx=dt\Rightarrow dx=2t\,dt,\] then it reduces to \[\int_{{}}^{{}}{2t\,.\cos t\,dt}=2\left[ t\,.\,\sin t-\int_{{}}^{{}}{\sin t\,dt} \right]\] \[=2t\sin t+2\cos t\]\[=2[\sqrt{x}\sin \sqrt{x}+\cos \sqrt{x}]+c\].
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