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INTEGRAL
EULERIANA |
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SOLUCIONES 4 |
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∫ |
π/2 |
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(2p-1=3 , 2q-1=0) |
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Γ(2)
· Γ(1/2) |
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Γ(1/2) |
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2 |
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| 1) |
sen3x dx |
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1/2 · β(2,1/2) |
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= |
= |
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2 · Γ(5/2) |
2 · 3/2 · 1/2 · Γ(1/2) |
3 |
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0 |
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∫ |
π/2 |
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(2p-1=3 , 2q-1=2/3) |
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Γ(2)
· Γ(5/6) |
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Γ(5/6) |
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36 |
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| 2) |
2 sen3x cos2/3x dx |
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β(2,5/6) |
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= |
= |
= |
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Γ(17/6) |
11/6 · 5/6 · Γ(5/6) |
55 |
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0 |
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∫ |
π/2 |
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∫ |
π/2 |
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(2p-1=-1/2 , 2q-1=1/2) |
Γ(1/4)
· Γ(3/4) |
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| 3) |
√(1/tg x) |
dx |
= |
sen-1/2x cos1/2x dx |
= |
1/2·β(1/4,3/4) |
= |
= |
1/2·Γ(1/4)·Γ(3/4) |
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| 2 ·
Γ(1) |
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0 |
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0 |
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∫ |
1 |
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Cambio Var. |
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∫ |
1 |
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(p-1=3 , q-1=-1/2) |
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4·Γ(4)·Γ(1/2) |
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128 |
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| 4) |
(1-x1/4)-1/2 |
dx |
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(x1/4= t) |
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4 |
t3 (1-t)-1/2 dt |
= |
4·β(4,1/2) |
= |
= |
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Γ(9/2) |
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35 |
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0 |
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0 |
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∫ |
∞ |
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1-√x |
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Cambio Var. |
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∫ |
∞ |
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(p-1=3/2) |
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3·e·√π |
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| 5) |
x1/4 |
e |
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(√x = t) |
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2·e |
e-t t3/2 dt |
= |
2·e·Γ(5/2) |
= |
2·e·3/2·1/2·Γ(1/2) |
= |
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2 |
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0 |
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0 |
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∫ |
∞ |
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(p-1=-2/3) |
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| 6) |
e-x / x2/3 |
dx |
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= |
Γ(1/3) |
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0 |
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∫ |
∞ |
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-x/4 |
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Cambio Var. |
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∫ |
∞ |
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(p-1=3) |
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| 7) |
x3 |
e |
dx |
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(x/4 = t) |
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44 |
e-t t3 dt |
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44 · Γ(4) |
= |
1.536 |
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0 |
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0 |
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∫ |
1 |
L x |
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Cambio Var. |
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∫ |
∞ |
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(p-1=1) |
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| 8) |
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dx |
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(x = e-2 t) |
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2 |
e-t t
dt |
= |
2 · Γ(2) |
= |
4 |
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√x |
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0 |
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0 |
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∫ |
1 |
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Cambio Var. |
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∫ |
∞ |
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(p-1=4) |
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| 9) |
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L4x |
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dx |
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(x = e- t) |
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e-t t 4 dt |
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Γ(5) |
= |
4! |
= |
24 |
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0 |
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0 |
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∫ |
1 |
x |
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∫ |
1 |
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(p-1=1/2 , q-1=-1/2) |
Γ(3/2)
· Γ(1/2) |
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π |
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| 10) |
√( |
) |
dx |
= |
x1/2 (1-x)-1/2 dx |
= |
β(3/2,1/2) |
= |
= |
1/2 · Γ2(1/2) |
= |
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| 1-x |
Γ(2) |
2 |
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0 |
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0 |
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∫ |
1 |
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dx |
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Cambio Var. |
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1 |
∫ |
1 |
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(p-1=-1/2 , q-1=-1/3) |
Γ(1/2)·Γ(2/3) |
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3√π·Γ(2/3) |
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| 11) |
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(x2 = t) |
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t-1/2(1-t)-1/3dt |
= |
1/2·β(1/2,2/3) |
= |
= |
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| 3√(1-x2) |
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2 |
2 · Γ(7/6) |
Γ(1/6) |
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0 |
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0 |
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∫ |
3 |
dx |
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Cambio Var. |
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∫ |
1 |
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(p-1=-1/2 , q-1=-1/2) |
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| 12) |
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(3-x = t) |
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t-1/2(1-t)-1/2dt |
= |
β(1/2,1/2) |
= |
π |
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2 |
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0 |
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∫ |
1 |
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dx |
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Cambio Var. |
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1 |
∫ |
1 |
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(p-1=-3/4 , q-1=-1/2) |
Γ(1/4)·Γ(1/2) |
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√π·Γ(1/4) |
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| 13) |
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(x2 = t) |
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t-3/4(1-t)-1/2dt |
= |
1/2·β(1/4,1/2) |
= |
= |
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2 |
2 · Γ(3/4) |
2·Γ(3/4) |
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0 |
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0 |
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∫ |
π/2 |
cos
x dx |
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Cambio Var. |
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| 14) |
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(sen x = t) |
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| √(1-sen
x) |
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0 |
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