求定积分∫4cos∧4θdθ上限π/2下限-π/2∫x∧4sinxdx上限π下限-π

Question

求定积分∫4cos∧4θdθ上限π/2下限-π/2 ∫x∧4sinxdx上限π下限 -π

Answer 1

第二题结果是0,因为被积函数是奇函数,积分区间对称,因此结果为0.

第一题,被积函数是偶函数：
∫[-π/2→π/2] 4(cosθ)^4 dθ
=8∫[0→π/2] (cosθ)^4 dθ
=2∫[0→π/2] (1+cos2θ)² dθ
=2∫[0→π/2] (1+2cos2θ+cos²2θ) dθ
=2∫[0→π/2] [1+2cos2θ+(1/2)(1+cos4θ)] dθ
=∫[0→π/2] (3+4cos2θ+cos4θ) dθ
=3θ + 2sin2θ + (1/4)sin4θ [0→π/2]
=3π/2

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