Question Number 114107 by mathdave last updated on 17/Sep/20

prove that  ∫_0 ^(π/2) [((ln(((1−sinx)/(1+sinx)))(√(cosx)))/((1+sinx)(√(1−sinx))))]dx=−8

Commented bymathdave last updated on 17/Sep/20

Commented byTawa11 last updated on 06/Sep/21

great sir

Answered by maths mind last updated on 17/Sep/20

=∫_0 ^(π/2) ((ln(((1−cos(x))/(1+cos(x))))(√(sin(x))))/((1+cos(x))(√(1−cos(x)))))dx  1+cos(x)=2cos^2 ((x/2)) and 1−cos(x)=2sin^2 ((x/2))⇒  =∫_0 ^(π/2) ((ln(tg^2 ((x/2)))(√(2sin((x/2))cos((x/2)))))/(2cos^2 ((x/2))(√(2sin^2 ((x/2))))))dx  =2∫_0 ^(π/2) ((ln(tg((x/2))))/( (√((sin((x/2)))/(cos((x/2)))))))(dx/(2cos^2 ((x/2))))  tg((x/2))=t⇒dt=(dx/(2cos^2 ((x/2)))) we get  =2∫_0 ^1 ((ln(x))/( (√x)))dx=[4(√x)ln(x)]_0 ^1 −4∫_0 ^1 (dx/( (√x)))=−4[2(√x)]_0 ^1 =−8