∫xcos⁡$$x+12x3esin$$⁡$$x+x2dx$$

Correct option is 1log ⁡2xesin⁡x+1−12xesin⁡x+1+1+C

Questions

$\int \frac{xcos x + 1}{\sqrt{2 x^{3} e^{\sin x} + x^{2}}} dx$

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log ⁡2xesin⁡x+1−12xesin⁡x+1+1+C

log ⁡2xesin⁡x−1+12xesin⁡x+1+1+C

log⁡2xesin⁡x+1+12xesin⁡x−1+1+C

log⁡2xesin⁡x+1+12xesin⁡x−1−1+C

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detailed solution

Correct option is A

I=∫(xcos⁡x+1)esin⁡xdxesin⁡xx2xesin⁡x+1 Put 2xesin⁡x+1=t2⇒ 2esin⁡x+esin⁡xcos⁡xxdx=2tdt⇒ esin⁡x(1+xcos⁡x)dx=tdt∴ I=∫tdtt2−12⋅t I=2∫dtt2−1=log⁡t−1t+1+C,where t2=2xesin⁡x+1

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∫xcos⁡x+12x3esin⁡x+x2dx

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$\int \frac{xcos x + 1}{\sqrt{2 x^{3} e^{\sin x} + x^{2}}} dx$