 | ∫{(sinθ)^2/(cosθ)^3}dθ
=∫{1/(cosθ)^3-1/cosθ}dθ
=∫{1/(cosθ)^4-1/(cosθ)^2}cosθdθ
ここでsinθ=tと置くとcosθdθ=dt
∴∫{(sinθ)^2/(cosθ)^3}dθ=∫[1/(1-t^2)^2-1/(1-t^2)}dt (A)
ここで更に
1/(1-t^2)^2=(at+b)/(1+t)^2+(ct+d)/(1-t)^2 (B)
と変形できるとすると
-a+b=lim[t→-1]1/(1-t)^2=1/4
c+d=lim[t→1]1/(1+t)^2=1/4
∴(B)は
1/(1-t^2)^2=(at+a+1/4)/(1+t)^2+(ct-c+1/4)/(1-t)^2 (B)'
∴
1/(1-t^2)^2={a(t+1)(t-1)^2+(1/4)(t-1)^2+c(t-1)(t+1)^2+(1/4)(t+1)^2}/(1-t^2)^2
={(t^2-1){a(t-1)+c(t+1)}+(1/2)(t^2+1)}/(1-t^2)^2
={(c+a)t^3-(c+a)t+(c-a+1/2)t^2-(c-a-1/2)}/(1-t^2)^2
分子の係数を比較すると
c+a=0
c-a=-1/2
これより(a,c)=(1/4,-1/4)
∴(B)'は
1/(1-t^2)^2=(1/4)(t+2)/(1+t)^2-(1/4)(t-2)/(1-t)^2
=(1/4)/(1+t)-(1/4)/(t-1)+(1/4)/(1+t)^2+(1/4)/(t-1)^2
となるので(A)は
∫{(sinθ)^2/(cosθ)^3}dθ
=∫[(1/4)/(1+t)-(1/4)/(t-1)+(1/4)/(1+t)^2+(1/4)/(t-1)^2
-(1/2)/(1-t)-(1/2)/(1+t)}dt
=∫[-(1/4)/(1+t)+(1/4)/(t-1)+(1/4)/(1+t)^2+(1/4)/(t-1)^2}dt
=(1/4)log{(t+1)/(t-1)}-(1/4)/(t+1)-(1/4)/(t-1)+C
=(1/4)log{(sinθ+1)/(sinθ-1)}-(1/4)/(sinθ+1)-(1/4)/(sinθ-1)+C
(C:積分定数)
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