Correct Answer - Option 1 : π/4
Formula used:
Definite integral:
∫ sin x = - cos x + C
∫ cos x = - sin x + C
Properties of definite integral:
\(\int_{a}^{b}f(x)dx\ =\ \int_{a}^{b}f(a\ +\ b\ -\ x)dx\)
Complementary angle formula:
\(sin\ (\dfrac{\pi}{2}\ -\ \theta)\ =\ cos \ \theta \)
\(cos\ (\dfrac{\pi}{2}\ -\ \theta)\ =\ sin \ \theta \)
Calculation:
\(I=\displaystyle∫_0^{π/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx\) ---(i)
\(I=\displaystyle∫_0^{π/2} \frac{\sqrt{\sin (π/2-x)dx}}{\sqrt{\sin (π/2-x)}+\sqrt{\cos (π/2-x)}}\)
\(I=\displaystyle∫_0^{π/2} \frac{\sqrt{\cos x}}{\sqrt{\cos x}+\sqrt{\sin x}}dx\) ---(iii)
Adding equations (1) and (3), we get
\(2I=\displaystyle∫_0^{π/2}\left(\frac{\sqrt{\sin x}+\sqrt{\cos x}} {\sqrt{\sin x}+\cos x}\right)dx\)
\(2I=[x]_0^{π/2}\)
⇒ I = π/4
Hence, option (1) is correct.
