Calcule:
(a) integral e^sen x cos x d x
(g) integral e^3 z cos (2 z) d z
(b) integral e^t1+e^2 t d t
(h) integral x^3 tg(x^4) d x
(c) integral 1raizx+1 d x
(i) integral sen x cos (cos x) d x
(d) integral sen^2 theta d theta
(j) integral raizcos (3 t) sen(3 t) d t
(e) integral 1y^2+6 y+8 d y
(k) integral(2 x+1) e^x^2 e^x d x
(f) integral xx^2+6 x+10 d x
(l) integral raiz1+raizx d x.

Passo 1

O enunciado pede doze integrais distintas. Vamos usar principalmente:
  • substituição u=f(x) quando aparecer f'(x) multiplicando a parte principal;
  • identidade trigonométrica sin^2theta=(1-cos2theta)/2;
  • frações parciais para funções racionais;
  • fórmula padrão
integral e^azcos(bz),dz =e^aza^2+b^2l(acos(bz)+bsin(bz)r)+C

Passo 2

Integrais do tipo integral f'(x),g(f(x)),dx:
(a) integral e^sin xcos x,dx
Defina u=sin x, du=cos x,dx.
integral e^sin xcos x,dx =integral e^u,du =e^u +C =e^sin x+C
(h) integral x^3tan(x^4),dx
Defina u=x^4, du=4x^3,dx então x^3,dx=tfracdu4.
integral x^3tan(x^4),dx =14integraltan u,du =-14lnl|cos ur|+C =-tfrac14lnl|cos(x^4)r|+C
(i) integral sin xcos(cos x),dx
Defina u=cos x, du=-sin x,dx.
integral sin xcos(cos x),dx =-integralcos(u),du =-sin u +C =-sinl(cos xr)+C
(j) integral raizcos(3t),sin(3t),dt
Defina u=cos(3t), du=-3sin(3t),dt então sin(3t),dt=-tfracdu3.
integral raizcos(3t),sin(3t),dt =-13integral u^1/2,du =-13cdot23u^3/2+C =-tfrac29,u^3/2+C =-tfrac29cos^3/2(3t)+C
(k) integral(2x+1)e^x^2e^x,dx
Note que x^2+x tem derivada 2x+1.
Defina u=x^2+x, du=(2x+1),dx.
integral(2x+1)e^x^2e^x,dx =integral e^u,du =e^u +C =e^x^2+x+C

Passo 3

Funções racionais e potências:
(b) integral e^t1+e^2t,dt
Defina u=e^t, du=e^t,dt.
integral e^t1+e^2t,dt =integraldu1+u^2 =arctan(u)+C =arctanl(e^tr)+C
(c) integral 1raizx+1,dx
Defina u=raizx, x=u^2, dx=2u,du.
integral dxraizx+1 =integral 2uu+1,du =integral2l(1-1u+1r),du =2u -2ln|u+1|+C =2raizx-2lnl(raizx+1r)+C
(d) integral sin^2theta,dtheta
Use sin^2theta=1-cos2theta2:
integral sin^2theta,dtheta =12integral dtheta-12integralcos2theta,dtheta =theta2-14sin2theta +C
(e) integral 1y^2+6y+8,dy
Fatore y^2+6y+8=(y+2)(y+4) e decomponha:
1(y+2)(y+4) =12l(1y+2-1y+4r)
Logo
integral dyy^2+6y+8 =12ln|y+2|-12ln|y+4|+C =12lnl|y+2y+4r|+C
(f) integral xx^2+6x+10,dx
Complete o quadrado: x^2+6x+10=(x+3)^2+1. Defina u=x+3, du=dx, x=u-3:
integral xx^2+6x+10,dx =integralu-3u^2+1,du =12lnl(u^2+1r)-3arctan(u)+C =12ln(x^2+6x+10)-3arctan(x+3)+C
(l) integral raiz1+raizx,dx
Defina u=raizx, dx=2u,du. Seja v=1+u, du=dv, u=v-1:
integral raiz1+raizx,dx =integral2uraiz1+u,du =2integral(v-1)v^1/2,dv =2integrall(v^3/2-v^1/2r),dv =2l(tfrac25v^5/2-tfrac23v^3/2r)+C =45v^5/2-43v^3/2+C =45(1+raizx)^5/2-43(1+raizx)^3/2+C

Passo 4

(g) integral e^3zcos(2z),dz
Use a fórmula com a=3, b=2:
integral e^3zcos(2z),dz =e^3z3^2+2^2l(3cos(2z)+2sin(2z)r)+C =e^3z13l(3cos(2z)+2sin(2z)r)+C

Resposta

(a) e^sin x+C
(b) arctan(e^t)+C
(c) 2raizx-2ln(raizx+1)+C
(d) tfractheta2-tfrac14sin(2theta)+C
(e) tfrac12ln!l|tfracy+2y+4r|+C
(f) tfrac12ln(x^2+6x+10)-3arctan(x+3)+C
(g) tfrace^3z13l(3cos(2z)+2sin(2z)r)+C
(h) -tfrac14lnl|cos(x^4)r|+C
(i) -sin(cos x)+C
(j) -tfrac29cos^3/2(3t)+C
(k) e^x^2+x+C
(l) tfrac45(1+raizx)^5/2-tfrac43(1+raizx)^3/2+C

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