Integration question (1 Viewer)

[ronnknee]

Evaluate the integral of:

sin x . cos x
--------------- dx
sin x + cos x


One obvious method is using t-substitution but it doesn't seem to work out that easily. Are there any other working methods that can be used for this?

 

[sicmacao]

Alternative method without t substitution

sinx cosx = 1/2 ( 2sinx cosx + sin^2x +cos^2x - 1) = 1/2 [(sinx +cosx)^2 - 1]

so

sin x . cos x
--------------- dx
sin x + cos x

= 1/2[ sinx + cosx - 1/(sinx + cosx) ] dx


now sinx + cosx = sqrt(2)[ cosx cos(pi/4) + sinx sin(pi/4) ] = sqrt(2) cos(x-pi/4)

hence 1/(sinx + cosx) = 1/sqrt(2) * sec(x-pi/4)

Hence

sin x . cos x
--------------- dx
sin x + cos x

= 1/2[ -cosx + sinx - 1/sqrt(2)* ln( sec(x-pi/4) + tan(x-pi/4) ) ]
Q.E.D.