Question

请问下图中的定积分要怎么计算？

如图，需要详细过程，谢谢

Answer 1

∫dt/ [(sint)^2.cost]
=∫ [(csct)^2.sect ] dt
=-∫ sect dcot(t)
=-sect.cot(t) + ∫ cot(t) .sect.tant dt
=-1/sint + ∫ sect dt
=-1/sint + ln|sect+tant| +C
∫( arctan(3/4)->arctan(4/3)) dt/ [(sint)^2.cost]

=[ -1/sint + ln|sect+tant| ] | ( arctan(3/4)->arctan(4/3))
=[ -5/4 + ln( 5/3 + 4/3) ] - [ -5/3 +ln(5/4+3/4) ]
=5/12 + ln3 - ln2
=5/12 + ln(3/2)