∫ (sinx)^6 dx

=∫ [(sinx)^2]^3 dx

=(1/8)∫ [ 1- cos2x ]^3 dx

= (1/8)∫ [ 1- 3cos2x + 3(cos2x)^2 - (cos2x)^3 ]dx

=(1/8)[ x - (3/2)sin2x] +(3/8)∫ (cos2x)^2 dx-(1/8)∫ (cos2x)^3 dx

=(1/8)[ x- (3/2)sin2x ]+(3/16)∫ (1+cos4x) dx-(1/16)∫ (cos2x)^2 dsin2x

=(1/8)[ x- (3/2)sin2x ] +(3/16)[x+(1/4)sin4x] -(1/16)∫[1- (sin2x)^2] dsin2x

=(1/8)[ x - (3/2)sin2x ] +(3/16)[x+(1/4)sin4x]-(1/16)[sin2x- (1/3)(sin2x)^3] +C

連續函式，一定存在定積分和不定積分；若在有限區間[a,b]上只有有限個間斷點且函式有界，則定積分存在；若有跳躍、可去、無窮間斷點，則原函式一定不存在，即不定積分一定不存在。