∫x3cos(x2) dx=∫x2cos(x2).xdx Put x2=t⇒2xdx=dt=12∫t.cost.dt=12t.sint−∫1−sint dt=12[t.sint+cost]+c=12xsinx+cosx+c=12x2sinx2+cosx2+c

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$\int x^{3} \cos (x^{2}) d x =$

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$\frac{1}{2} [x^{2} \sin (x^{2}) - \cos (x^{2})] + c$

$\frac{1}{3} [x^{2} \sin (x^{2}) + \cos (x^{2})] + c$

$\frac{1}{2} [x^{2} \sin (x^{2}) + \cos (x^{2})] + c$

$\frac{1}{3} [x^{2} \sin (x^{2}) - \cos (x^{2})] + c$

answer is C.

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Detailed Solution

$\int x^{3} \cos (x^{2}) d x$

$= \int x^{2} \cos (x^{2}) . x d x P u t x^{2} = t \Rightarrow 2 x d x = d t$

$= \frac{1}{2} \int t . \cos t . d t$

$= \frac{1}{2} [t . \sin t - \int 1 - \sin t d t]$

$= \frac{1}{2} [t . \sin t + \cos t] + c$

$= \frac{1}{2} [\sqrt{x} \sin \sqrt{x} + \cos \sqrt{x}] + c$

$= \frac{1}{2} [x^{2} {sinx}^{2} + {cosx}^{2}] + c$

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