_0^a x^2 x^3 ,dx equals - Vidyadip

MCQ

$\int_0^a {{x^2}\sin {x^3}\,dx} $ equals

A
$(1 - \cos {a^3})$
B
$3(1 - \cos {a^3})$
C
$ - \frac{1}{3}(1 - \cos {a^3})$
✓
$\frac{1}{3}(1 - \cos {a^3})$

✓

Answer

Correct option: D.

$\frac{1}{3}(1 - \cos {a^3})$

d
(d) $I = \int_0^a {{x^2}\sin {x^3}dx} $; Put ${x^3} = t $

$\Rightarrow {x^2}dx = \frac{{dt}}{3}$

$\therefore \,\,\,I = \frac{1}{3}\int_0^{{a^3}} {\sin t\,dt} $

$= - \frac{1}{3}[\cos t]_0^{{a^3}} = - \frac{1}{3}[\cos {a^3} - 1]$

$ = \frac{1}{3}[1 - \cos {a^3}]$.

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