求定积分:∫ln(tanx)dx(o≤x≤π/2),积分是限是π/2,下限是0

Question

求定积分:∫ln(tanx)dx (o≤x≤π/2),积分是限是π/2,下限是0

Answer 1

∫ln(tanx)dx
=∫[0,π/2] ln(tanx)dx
=∫[0,π/4]ln(tanx)dx+∫[π/4,π/2]ln(tanx)dx
=∫[0,π/4]ln(tanx)dx+∫[π/4,π/2]lncot(π/2-x)dx
=∫[0,π/4]ln(tanx)dx+∫[π/4,0]lncotud(π/2-u)
=∫[0,π/4]ln(tanx)dx+∫[π/4,0]lntanudu
=∫[0,π/4]ln(tanx)dx-∫[0,π/4]ln(tanu)du
=0