Question

sec的三次方如何用换元法求不定积分

Answer 1

∫sec³xdx
=∫secx·sec²xdx
=∫secx·(1+tan²x)dx
=∫secxdx+∫secxtan²xdx.......<这里观察到secxtanxdx=dsecx>
=ln|secx+tanx|+∫tanxdsecx
=ln|secx+tanx|+∫√(sec²x-1)dsecx
=ln|secx+tanx|+∫√(sec²x-1)dsecx
=ln|secx+tanx|+secx√(sec²x-1)/2-ln|secx+√(sec²x-1)|/2+c
盗来的...

Answer 2

S(secx)^3dx
=S(cosx/(1-(sinx)^2)dx
=S(1/(1-sinx)(1+sinx))dsinx
令：t=sinx
=S(1/(1+t)(1-t)dt
=1/2*S(1/(1+t)+1/(1-t))dt
=1/2*(ln(1+t)-ln(1-t)+c
=1/2*ln(1+t)/(1-t)+c
=1/2*ln(1+sinx)/(1-sinx)+c