(secx)^4(tanx)^5dx求不定积分,

(sex)^4(tanx)^5
=[(1/cosx)^4](sinx/cosx)^5
=(sinx)^5/(cosx)^9
=sinx[1-(cosx)^2]^2/(cosx)^9
=sinx[1-2(cosx)^2+(cosx)^4]/(cosx)^9
=sinx[1/(cosx)^9 -2/(cosx)^7 +1/(cosx)^5]
∫secx)^4(tanx)^5dx
=∫sinx[1/(cosx)^9 -2/(cosx)^7 +1/(cosx)^5]dx
=-∫[1/(cosx)^9 -2/(cosx)^7 +1/(cosx)^5]d(cosx)
=-[(-1/8)/(cosx)^8 -[2/(-6)]/(cosx)^6 +(-1/4)/(cosx)^4] +C
=1/[8(cosx)^8] -1/[3(cosx)^6] +1/[4(cosx)^4] +C