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∵cos^5x是偶函数
∴∫(-π/2,π/2)cos^5xdx
=2∫(0,π/2)cos^5xdx
=2∫(0,π/2)cos^4xdsinx
=2∫(0,π/2)(1-sin^2x)^2dsinx
=2∫(0,π/2) (1-2sin^2x+sin^4x)dsinx
=2(sinx-2/3sin^3x+1/5sin^5x)|(0,π/2)
=2[sinπ/2-sin0-2/3sin^3(π/2)+2/3sin^3(0)+1/5sin^5(π/2)-1/5sin^5(0)]
=2(1-0-2/3+0+1/5-0)
=2×(15/15-10/15+3/15)
=16/15
∴∫(-π/2,π/2)cos^5xdx
=2∫(0,π/2)cos^5xdx
=2∫(0,π/2)cos^4xdsinx
=2∫(0,π/2)(1-sin^2x)^2dsinx
=2∫(0,π/2) (1-2sin^2x+sin^4x)dsinx
=2(sinx-2/3sin^3x+1/5sin^5x)|(0,π/2)
=2[sinπ/2-sin0-2/3sin^3(π/2)+2/3sin^3(0)+1/5sin^5(π/2)-1/5sin^5(0)]
=2(1-0-2/3+0+1/5-0)
=2×(15/15-10/15+3/15)
=16/15
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