Doctorado

6. 15-4x7x-x2= 15-4x (7x-x2)-12d
u=15-4x du=-4 dv=(7x-x2)-12 v=2(7x-x2)12=(15-4x)∙(27x-x2)12-2(7x-4x2)12∙-4dx
=30-8x(7x-x2)12+8(7x-4x2)12dx
=30-8x(7x-x2)12 +8 23(7x-4x2)32+c
=30-8x(7x-x2)12 +163(7x-4x2)32+c

7. u=2xv=π02(fx-k 2 - gx-k 2 )dx y=k k=0
v=π02(2x-0 2 - x-0 2 )dx
v=π022x-x2dx
v=π2x22-x332 0 v=πx2-13x32 0v=π22-02-13(23-03) v=π4-83
v=π12-83 v=43π u3

8. v=π03(x+2-02 -x-02 )dx
v=π03x+22 -x2dxv=π03x2+4x+4-x2dx
v=π034x+4dx v=π4x22+4x3 0
v=π2x2+4x3 0 v=π2(32-02)+4(3-0)
v=π18+12 v=30πu3
9.A=2π34154y 1+dydx2 dx
A=2π34154x1+12x-122dx
A=2π34154x121+14x-1dx=2π34154x121+14xdxA=2π34154x124x+14xdx=2π34154x124x+12x12dx
A=2π12341544x+112dx
A= π341544x+112dx=π (4x+1)32432154 34
A= π6(4x+1)32 15434
A= π6415432-434+132A= π61632-432
A=π643-23 A= π6(64-8)
A= 283πu2
10. A=01(3-y-2y)dy A=013-y-2y12dyA=3y-y22-2y323210 A=3y-12y2-43y3210
A=31-0-1212-02-43132-032
A=3-12-43 A= 18-3-83 A=73u2
11. A= 044x-43x-43dx
A= 04(2x32-43x+43)dx
A=2 x3232- 43 x22+43 x40
A= 43 23- 23 16+163
A=323-323+163
A=16 3 u2