Definite Integrals — Maths STD 12 Science — Question

1. $2$

Solution:

$\int\limits^\frac{\pi^2}{4}_0\frac{\sin\sqrt{\text{x}}}{\sqrt{\text{x}}}\text{dx}$

Let $\sqrt{\text{x}}=\text{t},$ then $\frac{1}{2\sqrt{\text{x}}}\text{dx}=\text{dt}$

when $\text{x}=0,\text{t}=0,\text{x}=\frac{\pi^2}{4},\text{t}=\frac{\pi}{2}$

Therefore the integral becomes

$\int\limits^\frac{\pi}{2}_02\sin\text{t}\text{ dt}$

$=-2\big[\cos\text{t}\big]^\frac{\pi}{2}_0$

$=2$