Замена переменной:
х=10tgt
dx=10dt/cos^2t;
100+ x^2=100 +100tg^2t=100(1+ tg^2t)=100/cos^2t
sqrt((100+ x^2)^3)=sqrt(100^3/cos^6t)=1000/cos^3t
∫ dx/sqrt((100+ x^2)^3)= (1/100)*∫ costdt= [b](1/100)*(sint) C[/b]
x=10tgt ⇒ tgt=x/10
1 +tg^2t=1/cos^2t ⇒ cos^2t=1/(1+ tg^2t)=1/(1+(x/10)^2)=100/(100+ x^2)
sin^2t=1-cos^2t=1-(100/(100+ x^2))=x^2/(100+ x^2)
sint=x/sqrt(100+ x^2)
[b]О т в е т. ∫ dx/sqrt((100+ x^2)^3)= х/(100*sqrt(100 +x^2)) +C[/b]
2.
Замена переменной
x=t^4 ⇒ dx=4t^3dt
sqrt(x)=t^2
x^(1/4)=t
∫ (sqrt(x)-9)dx/(3x^(1/4) +sqrt(x))= ∫ (t^2-9)*4t^3dt/(3t+ t^2) =
= 4 ∫ (t^2*(t-3)(t+3)dt)/(t+3)=4 ∫ t^2*(t-3)dt=
=4 ∫ (t^3-3t^2)dt=4*((t^4/4)-4t^3 + C=
= [b]x-4x^(3/4)+C[/b]
О т в е т. [b] ∫ (sqrt(x)-9)dx/(3x^(1/4) +sqrt(x))=x-4x^(3/4)+C[/b]