定积分问题III

求∫[1/(cosx)^3]dx

∫[1/(cosx)^3]dx =∫(secx)^3dx =∫secx*(secx)^2dx =∫secxd(tanx) =secx*tanx-∫tanxd(secx) =secx*tanx-∫tanx*tanx*secxdx =secx*tanx-∫(tanx)^2*secxdx =secx*tanx-∫[(secx)^2-1]*secxdx =secx*tanx-∫(secx)^3dx+∫secxdx ===> 2∫(secx)^3dx=secx*tanx+∫secxdx ===> 2∫(secx)^3dx=secx*tanx+∫(1/cosx)dx ===> 2∫(secx)^3dx...全部

  ∫[1/(cosx)^3]dx =∫(secx)^3dx =∫secx*(secx)^2dx =∫secxd(tanx) =secx*tanx-∫tanxd(secx) =secx*tanx-∫tanx*tanx*secxdx =secx*tanx-∫(tanx)^2*secxdx =secx*tanx-∫[(secx)^2-1]*secxdx =secx*tanx-∫(secx)^3dx+∫secxdx ===> 2∫(secx)^3dx=secx*tanx+∫secxdx ===> 2∫(secx)^3dx=secx*tanx+∫(1/cosx)dx ===> 2∫(secx)^3dx=secx*tanx+∫(cosx/cos^2 x)dx ===> 2∫(secx)^3dx=secx*tanx+∫[1/(1-sin^2 x)]d(sinx) ===> 2∫(secx)^3dx=secx*tanx+(1/2)∫[(1/1-sinx)+(1/1+sinx)]d(sinx) ===> 2∫(secx)^3dx=secx*tanx+(1/2)ln|(1+sinx)/(1-sinx)|+C1 ===> ∫(secx)^3dx=(1/2)secx*tanx+(1/4)ln|(1+sinx)/(1-sinx)|+C。
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