The defined integral below is, in fact, very hard to evaluate by common means.
I = ∫oπ/2 √sin(x)/(√sin(x)+√cos(x)) dx
However, if you make the substitution x=(π/2-y), it becomes surprisingly easy to solve, by applying a basic concept of "defined integrals".
With this hint, can you, now, evaluate its value?
The required value of the definite integral is pi/4.
Substituting x=(pi -y), we obtain dx=dy so that:
I=integral(0 to pi/2)(Vsin(n/2-y)/(Vsin(n/2-y)+Vcos(n/2-y)) dy
= integral (0 to pi/2) Vcos y/(Vcos(y)+Vsin(y)) dy
= integral (0 to pi/2) Vcos x/(Vcos(x)+Vsin(x)) dx
(writing x for y)
= integral (0 to pi/2)(Vsin(x)+ Vcos(x)) /(Vsin(x)+Vcos(x)) dx
= integral (0 to pi/2) 1 dx
= pi/2, so that:
2I = pi/2
or, I = pi/4.
Consequently, the required value of the integral is pi/4
Edited on April 11, 2008, 1:55 pm