tg^34x=tg^24x * tg4x=(1-(1/cos^24x))*tg4x=tg4x-(tg4x/cos^24x)
∫ tg^34xdx= ∫ tg4x - ∫ tg4x/cos^24x=
= ∫ sin4xdx/cos4x - ∫ tg4x/cos^24x=
= ∫ (-(1/4)cos4x)/cos4x- ∫ tg4x (1/4)d(tg4x)=
= [b]-(1/4)ln|cos4x|-(1/8)*tg^24x + C[/b]
2.
Тригонометрические подстановки
х=2sint
dx=2costdt
4-x^2=4-4sin^2t=4(1-sin^2t)=4cos^2t
∫ sqrt((4-x^2)^3)dx/x^4= ∫ sqrt((4cos^2t)^3)*2costdt/(2sint)^4=
= ∫ cos^4tdt/sin^4t= ∫ ctg^4tdt= ∫ ctg^2t*ctg^2tdt=
= ∫ ctg^2t*(1 - (1/sin^2t))dt= ∫ ctg^2tdt - ∫ ctg^2tdt/sin^2t=
= ∫ (1-(1/sin^2t))dt - ∫ ctg^2tdt/sin^2t=
= ∫ dt - ∫ dt/sin^2t - ∫ ctg^2tdt/sin^2t=
= [b]t +ctg t - (ctg^3t)/3+C[/b]
3.
Раскладываем подынтегральную дробь на простейшие:
(5х+13)/(х+1)(x^2+6x+13)= A/(x+1) + (Mx+N)/(x^2+6x+13)
5х+13=А*(x^2+6x+13) + (Mx+N)*(x+1)
5x+13=(A+M)x^2+(6A+M+N)x+13A+N
A+M=0
6A+M+N=5
13A+N=13
А=1
М=-1
N=0
∫ (5х+13)dx/(х+1)(x^2+6x+13)= ∫ dx/(x+1) +∫ (-x)dx/(x^2+6x+13) =
=ln|x+1|-(1/2) ∫ (2x+6- 6 )dx/(x^2+6x+13)
=ln|x+1|-(1/2) ∫ (2x+6)dx/(x^2+6x+13)+3∫ dx/((x+3)^2+4)=
= [b]ln|x+1|-(1/2)ln|x^2+6x+13| +(3/2)arctg((x+3)/2) + C[/b]