7.2 definite integral — ગણિત ધોરણ 12 સાયન્સ — Question

$\int_0^2 {\frac{{{x^3}}}{{{{({x^2} + 1)}^{3/2}}}}dx = \frac{1}{2}} \int_1^5 {\frac{{(t - 1)}}{{{t^{3/2}}}}dt = \frac{1}{2}\int_1^5 {[{t^{ - 1/2}} - {t^{ - 3/2}}]\,dt} } $

$ = \frac{1}{2}\left[ {2\sqrt t + 2\frac{1}{{\sqrt t }}} \right]_1^5 $

$= \frac{1}{2}\left[ {2\sqrt 5 + \frac{2}{{\sqrt 5 }} - 2 - 2} \right]$

$ = \left[ {\sqrt 5 + \frac{1}{{\sqrt 5 }} - 2} \right] $

$= \frac{{6 - 2\sqrt 5 }}{{\sqrt 5 }}$.