∫x2ex3cos⁡ex3dx is equal to

A
$\sin (e^{x^{3}}) + C$
B
$3 \sin (e^{x^{3}}) + C$
C
$\frac{1}{3} \sin (e^{x^{3}}) + C$
D
$e^{x} \sin (e^{x^{3}}) + C$

Solution:

We have,

$\begin{array}{l} I = \int x^{2} e^{x^{3}} \cos (e^{x^{3}}) d x = \frac{1}{3} \int \cos (e^{x^{3}}) 3 x^{2} e^{x^{3}} d x \\ \Rightarrow I = \frac{1}{3} \int \cos (e^{x^{3}}) d (e^{x^{3}}) = \frac{1}{3} \sin (e^{x^{3}}) + C \end{array}$

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