Question

∫sec^3(x)/tan²xdx怎么算呐？

Answer 1

原式=∫(1/cos^3x)*(cos^2x/sin^2x)dx

=∫1/cosxsin^2xdx
=∫secxcsc^2xdx
=∫secx(cot^2x+1)dx
=∫secxcot^2xdx+∫secxdx
=∫cosx/sin^2xdx+ln|secx+tanx|
=∫d(sinx)/sin^2x+ln|secx+tanx|
=-1/sinx+ln|secx+tanx|+C
=ln|secx+tanx|-cscx+C，其中C是任意常数

Answer 2

∫sec^3xdx/tan^2x
=∫secxdtanx/tan^2x
=-∫secxdcotx
=-secxcotx+∫cotxsecxtanxdx
=-secxcotx+∫secxdx
=-secxcotx+(1/2)ln[(1+sinx)/(1-sinx)]+c