First slide

Methods of integration

Question

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

Moderate

A

$3 \sin (e^{x^{3}}) + C$
B

$\sin (e^{x^{3}}) + C$
C

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

$- \frac{1}{3} \sin (e^{x^{3}}) + C$

Solution

$\int x^{2} e^{x^{3}} \cos (e^{x^{3}}) d x,$ put $e^{x^{3}} = t$ $\Rightarrow e^{x^{3}} .3 x^{2} = \frac{d t}{d x}$
$= \frac{1}{3} \int 3 x^{2} e^{x^{3}} \cos (e^{x^{3}}) d x = \frac{1}{3} \int \cos t d t$

$= \frac{1}{3} (\sin t) + C .$

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