When X = Pi/4, Sin Pi/4 = Cos Pi/4. The Cosine Fun

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. And same thing happens in x = -pi/2 to x = 0