求解∫a^4cos^4xdx 积分上限为pai/2 下限为-pai/2

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求解∫a^4cos^4xdx 积分上限为pai/2 下限为-pai/2
求解∫a^4cos^4xdx 积分上限为pai/2 下限为-pai/2

求解∫a^4cos^4xdx 积分上限为pai/2 下限为-pai/2
∫(- π/2,π/2) a⁴cos⁴x dx
= 2a⁴∫(0,π/2) (cos²x)² dx,偶函数
= 2a⁴∫(0,π/2) [(1 + cos2x)/2]² dx
= 2a⁴∫(0,π/2) (1/4)(1 + 2cos2x + cos²2x) dx
= (a⁴/2)∫(0,π/2) (1 + 2cos2x) dx + (a⁴/2)∫(0,π/2) (1/2)(1 + cos4x) dx
= (a⁴/2)[x + sin2x] |(0,π/2) + (a⁴/4)[x + (1/4)sin4x] |(0,π/2)
= (a⁴/2)(π/2) + (a⁴/4)(π/2)
= (3/8)πa⁴