高数不定积分 高等数学 不定积分？
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不分积分法，如图
∫[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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