Integrales

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UNIVERSIDAD DE CUENCA
FACULTAD DE CIENCIAS ECONOMICAS Y ADMINISTRATIVAS
ASIGNATURA: MATEMATICAS




TEMA:

Tipos de Crédito y Tasas de Interés


PROFESOR:
Eco. Catalina Rivera


NOMBRES:





CURSO:
05-01

INTEGRALES

Fórmulas
y=fx dy=y’∙dx
u= ∅x du=u’∙dx
v=∅x dv=v’∙dx

∫dx=x+c
∫du+dv-dw=∫du+∫dv-∫dw
∫k∙dx=k∫dx∫xm∙dx=xm+1m+1+c
∫vm∙dv=vm+1m+1+c
∫1v’∙dv=∫dvv=∫v-1∙dv=Inv+c

Ejercicios

1. ∫4x35∙dx=4∫x35∙dx= 4x8585+c=45x858+4c=
5x852+c=4x8585+c=5x852+c

2. ∫2(2x)12∙dx=∫(2x)12 2∙dx=
v=2xdv=2∙2x⟹si está completo⟶∫vm∙dv
2∫2x∙dx=22∫x12dx=22∙2x323+c=42x323+c
3. ∫(x2-4)12∙x∙dx=

v=x2-4dv=2x∙dx=x.dx

y=2(x2+5)32+c=212(x2+5)12(2x)∙dx=6(x2+5)12 (x)∙dx

4. ∫6(x2+5)12x∙dx=

v=x2+5dv=2x∙dx=(x∙dx)∫(x2+5)12 6x∙dx=3∫(x2+5)12 2x∙dx=3(x2+5)2232+c=
2(x2+5)32+c

5. ∫12 (x2-4)-122x∙dx

v=x2-4dv=2x∙dx=x∙dx

12∫(x2-4)-12 2x∙dx= 12x2-412+c=(x2-4)12+c

6. 154∫3x2-52∙dx=

v=3x2-5dv=32∙dx≠dx
52∫3x2-52∙32∙dx= 523x2-53+c= 563x2-53+c

7. ∫x2-52∙dx=

v=x2-5dv=2x∙dx≠2∙dx

2∫x4-10x2+25∙dx=2x55-10x33+25x+c

8. ∫(2x-7)2∙dx=

v=2x-7dv=2∙dx≠dx

12∫(2x-7)2 2∙dx=12(2x-7)33+c= (2x-7)36+c

9. ∫(5x-6)13 32∙dx=

v=5x-6dv=5∙dx=32∙dx

3215∫(5x-6)13 5∙dx= 325(5x-6)4343+c=245(5x-6)43+c

10. 154∫9x24-15x+25∙dx=

1549x343-15x22+25x+c

11. ∫a-xa-v∙dx=

∫a12∙dx+∫x12∙dx= a∙x+2x323+c

12. 42∫(2x+3)12 42∙dx=

v=2x+3dv=2∙dx=dx

21∫(2x+3)122∙dx=21(2x+3)3232+c= 14(2x+3)32+c

13. ∫2x+3∙dxx+2=

v=x+2dv=dx=dx∫2-1x+2∙dx=2∫dx-∫dxx+2=2x-lnx+2+c

14. ∫x2+2x+1∙dx=

∫x-1+3x+1∙dx= x22-x+3∫dxx+1=x22-x+3lnx+1+c

15. ∫ex∙dx(a+bex)=

v=a+bexdv=b∙ex∙dx≠ex∙dx→no esta completo

1b∫bex∙dxa+bex= 1bln⁡(a+bex)+c

16. ∫(x2+1)∙dxx3+3x=

v=a+bexdv=3x2+5∙dx=3x2+1∙dx

∫x2+1∙dx(x3+3x)12= 13∫(x2+3x)123x2+1∙dx=(x3+3x)122+c=

2x3+3x3+c

17. ∫e2s∙ds(e2s+1)12=

v=e2s+1dv=2e2s∙dx≠e2s

12∫2e2s∙dxe2s+1=12lne2s+1+c18. ∫aex+b∙dxaex-b=

v=aex-bdv=aex∙dx=aex∙dx


b+aex
-b+aex
‖+2aex

∫2aexaex-b-∫dx=2∫aexaex-b-x=2∫lnaex-b-x+c

19.3t dt= 313dt=313t13dt=313t13+113+1+C=313t4343+C=3133t434.+C
= 34/3t4/34+C

20.a+bx dx= a+bx12dx=1ba+bx12b.dx=1ba+bx12+112+1+C
=1ba+bx3/23/2+C= 2a+bx3/23b+C

21.x 2+x2dx=12 2+x222xdx=12 2+x22+12+1+C= 12 2+x233+C
= 2+x236+C22.x22-2x2 dx=12 x2dx-12-1x-2dx=12 x2+12+1-12-1x-2+1-2+1+C
= 12 x33-12-1 x-1-1+C= x36+2x+C

23.xn-1 a+bxn dx= xn-1 a+bxn12 dx=1bna+bxn12 bn xn-1 dx
= 1bna+bxn12+112+1+C= 2a+bxn3/23bn+ C

24.y+2y2+4y dy= 22y+2y2+4y dy= 122y+4y2+4y dy= lny2+4y2+ C

25.eϑdϑa+beϑ= bbeϑdϑa+beϑ= 1bbeϑdϑa+beϑ= lna+beϑb+ C

26.2xdx36-5x2 =6-5x2-13 2xdx= -15 6-5x2-13 .-10xdx
= -15 6-5x2-13+1-13+1+C=-15 6-5x22323 + C= -15 36-5x2232 + C
=-310 6-5x223+C

27.t 2t2+3 dt= 142t2+312 4tdt= 14 2t2+312+112+1+ C
= 14 2t2+33/23/2+ C= 22t2+33/24 . 3+ C= 2t2+33/26+ C

28.x a-x2dx= x12a-2a12+xdx= ax12-2a12x12+x32dx=
ax12 .dx-2a12 x .dx+x32 dx=a x12+132-2a12 x1+11+1+x32+132+1+C=
2ax3/23-a1/2x2+2x5/25+C

Gráficos.- Integrales

1. y=x

k=-22x∙dx=x2+c-22
k=222+c-(-2)22+c
k=2+c-2-c=0k=-20x∙dx+02x∙dx=∣-2∣+∣2∣=4u2

2. y=3x2

k=-203x2∙dx=3x33+c-20

k= 0+C- -8+C

k=8 u2

3. y=x2
y=4-x
y=x

x=2 y=2

k=01x2dx+12xdx+244-xdx

k=13+x22+C12+4x-x22+C2

k= 13+32+2=3.83u2

4. y=x
y=x2




0=x2-x

0=xx-1

X=0 x-1=0

x=1

k=Fb-Fak=01xdx-01x2dx

k=x22+C01-x33+C01


k=12+C-13-C=16u2



5. y=x
y=x2



k=01x2dx-13xdx


k=x33+C01-x22+C13

k=13+ C+92-12+ C

k=13+ C+4+C=13+4=133 u2

6. y=x2
y=4-x (0; 4)
y=x (4 ;0)

y=x2
y=4-x
0=4+x+x2
x=-1±1+162=1+482 x1=1.56 y x2=-2.56
k=-2.561.564-xdx--2.560x2dx+11.56x2dx...
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