Question

Find the definite integral,

∫ (from pi/2 to pi) sin(x)/1+cos^2(x)

Answer 1

Calculate the ∫ (from pi/2 to pi) sin(x)/1+cos^2(x).

First find the ∫ sin(x)dx/1+cos^2(x).

Let cosx =t

dt/dx =-sinx

-dt =sinx dx

∫ sin(x)dx/1+cos^2(x). =∫ -dt/1+t^2

                                 =-tan^-1(t)

                                 =-tan^-1(cosx)

Calculate the definite integration using the fundamental theorem of intehration.

∫ (from pi/2 to pi) sin(x)/1+cos^2(x).=(from pi/2 to pi)( -tan^-1(cosx))

                                                     =-[tan^-1(cospi/2)-tan^-1(cospi)]

                                                     =-[0-(tan^-1(-1)]

                                                     =-(-pi/4)

                                                     = pi/4