求定积分简便算法，高等数学

Question

如题所述

Answer 1

0到2π的偶函数积分为0

Answer 2

题目不完整追问

只要最后一步计算

追答

你写错了。对 t 积分上限是 2π。
被积函数
(1-cost)^4 = 1 - 4cost + 6(cost)^2 - 4(cost)^3 + (cost)^4
= 1 - 4cost + 3(1+cos2t) - 4(cost)^3 + (1/4)(1 + cos2t)^2
= 1 - 4cost + 3 + 3cos2t - 4(cost)^3 + (1/4)[1 + 2cos2t + (cos2t)^2]
= 1 - 4cost + 3 + 3cos2t - 4(cost)^3 + 1/4 + (1/2)cos2t
+ (1/8)(1+cos4t)
= 35/8 - 4cost + (7/2)cos2t + (1/8)(cos4t) - 4(cost)^3
则积分
I = (a^4/3) ∫ [35/8 - 4cost + (7/2)cos2t
+ (1/8)(cos4t) - 4(cost)^3] dt
= (a^4/3) [35t/8 - 4sint + (7/4)sin2t + (1/32)sin4t
- 4{sint - (sint)^3/3}]
= (35/12)πa^4