∫ x*(sinx)^6(cosx)^4在0到pi上的定积分

Question

∫ x*(sinx)^6(cosx)^4在0到pi上的定积分

Answer 1

根据公式：
∫(0,π)xf(sinx)dx
=π/2∫(0,π)f(sinx)dx
可得
原式
=π/2 ∫(0,π)sin^6xcos^4xdx
=π∫(0,π/2)sin^6x[1-sin^2x]^2dx
=π∫(0,π/2)sin^6x[1-2sin^2x+sin^4x]dx
=π∫(0,π/2)[sin^6x-2sin^8x+sin^10x]dx
=π×【5/6×3/4×1/2×π/2 -2×7/8×5/6×3/4×1/2×π/2+9/10×7/8×5/6×3/4×1/2×π/2】
=π×3π/512
=3π方/512