Question Number 101192 by bemath last updated on 01/Jul/20

∫ (x/(1+sin x)) dx

Commented bybobhans last updated on 01/Jul/20

set z = (π/2)−x ⇒ dx = −dz I=∫ (((π/2)−z)/(1+sin ((π/2)−z))) (−dz) = ∫ ((z−(π/2))/(1+cos z)) dz = ∫ (z−(π/2)) d(tan ((z/2))) = (z−(π/2))tan ((z/2))+2 ln(cos ((z/2))) + c = −x tan (((π−2x)/4)) + 2ln (cos (((π−2x)/4))) + c

Commented bybemath last updated on 01/Jul/20

thank you both

Answered by smridha last updated on 01/Jul/20

∫((x(1−sinx))/((1+sinx)(1−sinx)))dx =∫xsec^2 x dx−∫xtanx.secx dx =xtanx+ln[cosx]−xsecx+ln[tan{(𝛑/4)+(x/2)}]+c or =x(tanx−secx)−ln(secx)+ln[secx+tanx]+c