sin^5x=sin^4x*sinx=(sin^2x)^2*sinx=(1-cos^2x)^2*sinx
Замена переменной
cosx=t
dt=(cosx)`*dx
dt=-sinxdx
sinxdx=-dt
получим
∫ (1-t^2)^2*(-dt)/(3-t)= ∫ (t^4-2t^2+1)dt/(t-3)
неправильная дробь. Выделяем целую часть.
= ∫ (t^3+3t^2+7t+21+ (64/(t-3))dt=
= [b](t^4/4)+(3t^3/3)+(7t^2/2)+21t+64ln|t-3|+C, t=cosx[/b]
2.
2sin^2x+9cos^2x=2cos^2x*(t^2+(9/2))
Замена переменной
tgx=t
dx/cos^2x=dt
получим:
1/2∫ dt/(t^2+(9/2))=
Формула ∫ dx/(x^2+a^2)=(1/a)* arctg (x/a); a^2=9/2 a=3/sqrt(2)
=(1/2)*(sqrt(2)/3)arctg( sqrt(2)*t/3) + C=
= [b](sqrt(2)/6)*arctg((sqrt(2)tgx)/3) + C[/b]
3.
Замена
tgx=t
x=arctg t
dx=dt/(1+t^2)
1+tg^2x=1/cos^2x;
cos^2x=1/(1+tg^2x)=1/(1+t^2)
sin^2x=tg^2x*cos^2x=t^2/(1+t^2)
sin^8x=(sin^2x)^4=t^8/(1+t^2)^4
cos^(14)x=1/(1+t^2)^7
sin^8x/cos^(14) =(t^8/(1+t^2)^4)* (1+t^2)^7= (1+t^2)^3*t^8
sin^8x dx /cos^(14)= (1+t^2)^3*t^8 * (dt/(1+t^2))
получим
∫ t^8*(1+t^2)^2dt= ∫ t^8*(1+2t^2+t^4)dt= ∫ (t^(12)+2t^(10)+t^8)dt=
=t^(13)/13 + 2t^(11)/11+ t^(9)/9 + C=
= [b](tgx)^(13)/13 + 2(tgx)^(11)/11 + (tgx)^(10)/10 + C[/b]
4.
sin^2x=(1-cos2x)/2;
cos^2x=(1+cos2x)/2
sin^2x*cos^2x=(1-cos2x)*(1+cos2x)/4=(1-cos^22x)/4=
=(1/4) - (1/4) cos^22x=(1/4) - (1/4)*(1+cos4x)/2=
=(1/4)-(1/8) -(1/8) cos4x= (1/8) -(1/8)cos4x
∫ sin^2x*cos^2xdx= ∫ ((1/8) -(1/8)cos4x)dx=
= [b](1/8)x - (1/32) * sin4x + C[/b]