∫ cos^6xdx= ∫ (cos^2x)^3dx= ∫ ((1+cos2x)/2)^3dx=
=∫ (1+3cos2x+3cos^32x+cos^32x)dx/8
= ∫ ((1/8)+(3/8)*cos2x+(3/8)*((1+cos4x)/2) + (1/8)*(1-sin^2x)*cos2x)dx
=((1/8)+(3/16) )x +(3/8)*(1/2)sin2x + (3/16)*(1/4)sin4x +(1/8)*(1/2)sin2x-
-(1/8)*(1/2)*(sin^32x)/3) + C
19б
sec^44x=1/cos^4(4x)=(1/cos^24x)*(1/cos^24x)=(tg^24x+1)*(1/cos^24x)
получаем
Замена
tg4x=t
dt=(tg4x)`dt=
dt= 4dx/cos^24x
dx/cos^24x=(1/4)dt
∫ (2-tg^34x)sec^44x= ∫ (2-t^3)*(t^2+1)*(1/4)dt=
=(1/4) ∫ (2t^2-t^5-t^3+2)dt=
=(1/4)*(2t^3/3) - (1/4)*(t^6/6) -(1/4)*(t^4/4)+(1/4)*2t+ C, t=tg4x