_ ^ x^3 e^ x^2 dx = - Vidyadip

MCQ

$\int_{}^{} {{x^3}{e^{{x^2}}}dx = } $

A
$\frac{1}{2}({x^2} + 1){e^{{x^2}}} + c$
B
$({x^2} + 1){e^{{x^2}}} + c$
✓
$\frac{1}{2}({x^2} - 1){e^{{x^2}}} + c$
D
$({x^2} - 1){e^{{x^2}}} + c$

✓

Answer

Correct option: C.

$\frac{1}{2}({x^2} - 1){e^{{x^2}}} + c$

c
(c) Put ${x^2} = t \Rightarrow 2x\,dx = dt,$ then $\int_{}^{} {{x^3}{e^{{x^2}}}dx} = \frac{1}{2}\int_{}^{} {t{e^t}dt} $
$ = \frac{1}{2}\left[ {t{e^t} - {e^t}} \right] + c$$ = \frac{1}{2}{e^{{x^2}}}({x^2} - 1) + c.$

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