Question

∫(sinx)^2/(cosx)^3 dx 求不定积分 我们只学了凑微分法和换元法,

Answer 1

∫ sin²x/cos³x dx
= ∫ tan²xsecx dx
= ∫ (sec²x - 1)secx dx
= ∫ sec³x dx - ∫ secx dx
对于∫ sec³x dx = ∫ secx d(tanx)
= secxtanx - ∫ tanx d(secx)
= secxtanx - ∫ tanx(secxtanx) dx
= secxtanx - ∫ (sec²x - 1)secx dx
= secxtanx - ∫ sec³x dx + ∫ secx dx
2∫ sec³x dx = secxtanx + ∫ secx dx
∫ sec³x dx = (1/2)secxtanx + (1/2)∫ secx dx
∫ sin²x/cos³x dx = (1/2)secxtanx + (1/2 - 1)∫ secx dx
= (1/2)secxtanx - (1/2)ln| secx + tanx | + C