Question

∫sec^3(x)/tan²xdx怎么算?

Answer 1

先对不定积分进行变形，得到：
∫sec^3(x)/tan²xdx
=∫[1/cos^3(x)]/[sin^2x/cos^2x]dx
=∫[1/cos(x)]/[sin^2x]dx
=∫1/[cos(x)*sin^2x]dx
=∫dsinx/[cos^2(x)*sin^2x]
=∫dsinx/[(1-sin^2(x))*sin^2x]
=∫dsinx/[(1-sin^2(x)]+∫dsinx/(sin^2(x))
=(1/2)ln[(1+sinx)/(1-sinx)]-(1/sinx)+c

Answer 2

原式=∫ (1/cos^3x)*(cos^2x/sin^2x) dx
=∫ 1/cosxsin^2x dx
=∫ secx*csc^2x dx
=∫ secx*(cot^2x+1) dx
=∫ (secx*cot^2x+secx) dx
=∫ (cotx*cscx+secx) dx
=-cscx+ln|secx+tanx|+C，其中C是任意常数