>>893
=∫[0,π/2]{(sint)^8/(sint*cost)}*2sintcost dt
=2∫[0,π/2] (sint)^8 dt

I(n)=∫[0,π/2](sint)^n dt とおくと
I(n)=[-cost(sint)^(n-1)][0,π/2]+(n-1)∫[0,π/2](cost)^2(sint)^(n-2)dt=(n-1){I(n-2)-I(n)}
よって　I(n)={(n-1)/n}I(n-2)
I(8)=(7/8)I(6)=(7/8)*(5/6)I(4)=(7/8)*(5/6)*(3/4)I(2)=(7/8)*(5/6)*(3/4)*(1/2)I(0)=(105/768)π

与式=2I(8)=105π/384