Calculate the definite integral of function ymaximumsin xcos

Calculate the definite integral of function y=maximum(sin x,cosx)?

limits of integration x=0 and x=pi/2

Solution

When x = pi/4, sin pi/4 = cos pi/4.

The cosine function is decreasing over the set [0,pi/2] and sine function is increasing over the set [0,pi/2].

cos x>sin x, if x belongs to the set [0,pi/4]

sin x>cos x, if x belongs to the set [pi/4,pi/2]

Therefore, the definite integral will split into 2 integrals:

Int maximum(sin x,cosx)dx = Int cos xdx (0->PI/4) + Int sin xdx (PI/4 -> PI/2)

Int cos xdx (0->PI/4) = sin x(0->PI/4)

Int cos xdx = sin pi/4 - sin 0

Int cos xdx = sqrt2/2

Int sin xdx (PI/4 -> PI/2) = -cos x(PI/4 -> PI/2)

Int sin xdx = -cos pi/2 + cos pi/4

Int sin xdx = sqrt2/2

Int maximum(sin x,cosx)dx = sqrt2/2 + sqrt2/2

Int maximum(sin x,cosx)dx = sqrt 2

The value of definite integral of function maximum(sin x,cosx), if the limits of integration are x = 0 to x = pi/2, is: Int maximum(sin x,cosx)dx = sqrt 2.