Question

∫sin²（根号x）dx

Answer 1

楼上都答得好乱，还是我的最清楚~~答题步骤清晰是一种美德，也是尊敬考官的行为。
令x=z²，dx=2z dz
∫sin²√x dx
= ∫(sin²z)(2z dz)
= 2∫z*(1-cos2z)/2 dz
= ∫z dz - ∫zcos2z dz
= ∫z dz - (1/2)∫z d(sin2z)，分部积分法
= ∫z dz - (1/2)zsin2z + (1/2)∫sin2z dz，分部积分法
= z²/2 - (1/2)zsin2z - (1/4)cos2z + C，回代z=√x
= x/2 - (1/2)(√x)sin(2√x) - (1/4)cos(2√x) + C

Answer 2

t=根号x
∫sin²（根号x）dx等价于 -1/2∫sin² t d t = - 1/2（t/2-1/4sin2t+C）=1/8sin2t-t/4+C

Answer 3

令t=√x则
∫sin²(√x)dx=-0.5∫sin² t d t=-0.5(t/2-1/4×sin2t+C)=0.125sin2t-t/4+C