7.2 definite integral — ગણિત ધોરણ 12 સાયન્સ — Question

Using the property $I = \int_a^b {f(x)dx = \int_a^b {f(a + b - x)} dx} $

$i.e.,$ change in $x = (2 + 3 - x) = 5 - x$ or $dx = - dx$

$\therefore I = \int_3^2 {\frac{{\sqrt {5 - x} }}{{\sqrt x + \sqrt {5 - x} }}} ( - dx)$

$ = \int_2^3 {\frac{{\sqrt {5 - x} }}{{\sqrt {5 - x} + \sqrt x }}dx} $ ....$(ii)$

Adding $(i)$ and $(ii),$

$2I = \int_2^3 {\frac{{\sqrt x + \sqrt {5 - x} }}{{\sqrt {5 - x} + \sqrt x }}dx = \int_2^3 {1dx} } $

$ = [x]_2^3 = 3 - 2 = 1 \Rightarrow I = \frac{1}{2}$.