Find the value of

$\int_1^{\sqrt{2}} x^3\sqrt{x^2-1}dx.$

- It is expected that you solve this in 2 minutes. (without calculator of course)

*
This problem is a part of
<Grade 12 CSAT Mock test> series
.
*

$\dfrac{11}{15}$
$\dfrac{2}{3}$
$\dfrac{3}{5}$
$\dfrac{8}{15}$
$\dfrac{7}{15}$

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Let $x^2-1=t,$ so that $2x dx = dt.$ Then,

$\int_1^{\sqrt{2}}x^3\sqrt{x^2-1}dx \\ =\dfrac{1}{2}\int_0^1 (t+1)\sqrt{t}dt \\ =\dfrac{1}{2}\left[\frac{2}{5}t^{\frac{5}{2}}+\frac{2}{3}t^{\frac{3}{2}}\right]_0^1 \\ =\boxed{\dfrac{8}{15}}.$