题目：一个固定的光滑的1/4圆形轨道，有一物体从轨道{zg}点静止下落，求物体从{zg}点到最低点所用的时间。

解：

设转过角度为θ，则角加速度β=gcosθ/r=dω/dt

∴dω/cosθ=gdt/r

∵角速度ω²=v²/r²=2grsinθ/r²=2gsinθ/r

∴ [d√(2gsinθ/r)]/cosa=gdt/r

∴ (d√sinθ)/cosθ=√(g/2r)dt

∴dθ/√sinθ=√(g/2r)dt

∴∫^π/2₀dθ/√sinθ=∫^T₀√(g/2r)dt

由积分表可查得∫dx/sinⁿx=-cosx/[(n-1)sin^n-1x]+(n-2)/(n-1)×∫dx/sin^n-2x

∴∫dx/sin^1/2x=2 sin^1/2xcosx+3∫dx/sin^-3/2x

=2 sin^1/2xcosx+(6/5) sin^5/2xcosx+(21/5) ∫dx/sin^-7/2x

=2 sin^1/2xcosx+(6/5) sin^5/2xcosx+(21/5)[(2/9) sin^9/2xcosx+(11/9) ∫dx/sin^-11/2x]

=    … …

代入积分上下限后除{zh1}一项外都是0

∴∫^π/2₀dx/√sinx=3×7/5×11/9×… …×(4n-1)/(4n-3) ×∫^π/2₀ sin^2n-1/2dx

∵∫^π/2₀ sinⁿxdx=∫^π/2₀sin^n-1xdcosx=[-cosxsin^n-1x] ^π/2₀+∫^π/2₀cosxdsin^n-1x

=(n-1) ∫^π/2₀ sin^n-2xcos²xdx=(n-1) ∫^π/2₀(sin^n-2x-sinⁿx)dx

=(n-1) ∫^π/2₀ sin^n-2xdx-(n-1) ∫^π/2₀ sinⁿxdx

∴令∫^π/2₀ sinⁿxdx=I_n ,则I_n=(n-1)I_n-2-(n-1)I_n

∴I_n=[(n-1)/n]×I_n-2

I_n-2=[(n-3)/(n-2)] ×I_n-4

… …

∴I_2n=(2n-1)/(2n) ×(2n-3)/(2n-2) ×(2n-5)/(2n-4) ×… …×5/6×3/4×1/2×I₀

   I_2n-1=(2n-2)/(2n-1) ×(2n-4)/(2n-3) ×(2n-6)/(2n-5) ×… …×6/7×4/5×2/3×I₁

∵I₀=∫^π/2₀dx=π/2    , I₁=∫^π/2₀ sinxdx=1

∴I_2n=∫^π/2₀ sin²ⁿxdx=(2n-1)/(2n) ×(2n-3)/(2n-2) ×(2n-5)/(2n-4) ×… …×5/6×3/4×1/2×π/2

   I_2n-1=∫^π/2₀ sin^2n-1xdx=(2n-2)/(2n-1) ×(2n-4)/(2n-3) ×(2n-6)/(2n-5) ×… …×6/7×4/5×2/3

∵min {∫^π/2₀ sin²ⁿxdx , ∫^π/2₀ sin^2n-1xdx }≦∫^π/2₀ sin^2n-1/2xdx≦max{∫^π/2₀ sin²ⁿxdx , ∫^π/2₀ sin^2n-1xdx }

而lim_(n→∞) 3×7/5×11/9×15/13×… …×(4n-1)/(4n-3) ×∫^π/2₀ sin²ⁿxdx

= lim_(n→∞) 3×7/5×11/9×15/13×… …×(4n-1)/(4n-3) ×(2n-1)/(2n) ×(2n-3)/(2n-2) ×… …×5/6×3/4×1/2×π/2

= lim_(n→∞)   3×1/2×7/5×3/4×11/9×5/6×15/13×7/8×… …×[(4n-1)(2n-1)]/[(4n-3)(2n)] ×π/2

≈2.6            

lim_(n→∞) 3×7/5×11/9×15/13×… …×(4n-1)/(4n-3) ×∫^π/2₀ sin^2n-1xdx

=lim_(n→∞) 3×7/5×11/9×15/13×… …×(4n-1)/(4n-3) ×(2n-2)/(2n-1) ×(2n-4)/(2n-3) ×… …×6/7×4/5×2/3

= lim_(n→∞) 3×2/3×7/5×4/5×11/9×6/7×… …×[(4n-1)(2n-2)]/[(4n-3)(2n-1)]

≈2.6

∴∫^π/2₀ dx/sin^1/2x=lim_(n→∞) 3×7/5×11/9×15/13×… …×(4n-1)/(4n-3) ×∫^π/2₀ sin^2n-1/2xdx

=lim_(n→∞) 3×7/5×11/9×15/13×… …×(4n-1)/(4n-3) ×∫^π/2₀ sin²ⁿxdx

= lim_(n→∞) 3×7/5×11/9×15/13×… …×(4n-1)/(4n-3) ×∫^π/2₀ sin^2n-1xdx

≈2.6

∴2.6≈T×√(g/2r)

∴T≈2.6×√(2r/g)