Задача 33281 ...

sova

2019-02-05 13:41:53

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1.
ctg^34x=ctg4x*(ctg^24x)=ctg4x*((1/sin^2x)-1)

d(ctg4x)=(ctg4x)`dx=(-1/sin^24x)*(4x)`=-4dx/sin^24x


∫ ctg^34xdx= ∫ ctg4x*dx(1/sin^2x)- ∫ ctg4x=

=(-1/4) ∫ ctg4x d(ctg4x)- ∫ cos4xdx/sin4x=

=(-1/4)(ctg^2(4x))/2-(1/4) ∫ d(sin4x)/sin4x=

(-1/8)ctg^2(4x)-(1/4)ln|sin4x|+С

2.

1-3cos^2x+sin^2x=sin^2x+cos^2x-3cos^2x+sin^2x=2sin^2x-2cos^2x=

= -2*cos2x

∫ dx/(-2cos2x)= ( - 1/4) ∫d(2x)/cos(2x)=

cм. таблицу

= ( -1/4) ln |tg x +(π/4)| + C

3.
Формула
sin α * cos β =(1/2)sin( α + β ) + (1/2) sin ( α - β )

sin 3x * cos 2x = (1/2)sin5x +(1/2) sinx

∫sin3x*cos2x=(1/2) ∫ sin5x dx +(1/2) ∫ sinx dx =

=(1/10)*(-cos5x)+(1/2)*(-cosx) + C