Trabajo: Cálculo de integrales indefinidas, no inmediatas

𝑏

𝑣 = ∫ 𝐴(𝑥)𝑑𝑥 → 𝑀𝐸𝑇𝑂𝐷𝑂 𝐷𝐸 𝐷𝐼𝑆𝐶𝑂𝑆

𝑎

Funciones:

[pic 1]

𝐴(𝑥) = 𝜋[𝑓(𝑥)]2 − 𝜋[𝑔(𝑥)]2 = 𝜋([𝑓(𝑥)]2 − [𝑔(𝑥)]2)

𝐴(𝑥) = 𝜋([2𝑥 𝑒

−𝑥]2

𝑥

− [𝑠𝑒𝑛 ([pic 2]

2

)] )

10

𝐴(𝑥) = 𝜋(4𝑥 𝑒−2𝑥        2 𝑥 )) 10[pic 3]

𝑉 = 𝜋 ∫ (4𝑥2 𝑒−2𝑥 − 𝑠𝑒𝑛2( 𝑥[pic 4][pic 5]

0        10

Definición de integración por partes:

))𝑑𝑥

∫ 𝑢𝑣1 = 𝑢𝑣 − ∫ 𝑢1𝑣

= [𝑥2 (− 1[pic 6][pic 7][pic 8]

2

𝑒−2𝑥) − ∫ 2𝑥 (− 1

2[pic 9]

2

𝑒−2𝑥) 𝑑𝑥]

0

Simplifico:

1

= 4 [−[pic 10]

2

2

𝑒−2𝑥𝑥2 − ∫ −𝑒−2𝑥𝑥𝑑𝑥]

0

= 4 [−

1 𝑒−2𝑥𝑥2 − (− 1

2        4[pic 11][pic 12]

2

(−2𝑒−2𝑥𝑥 − 𝑒−2𝑥))]

0

= 4 [−

1 𝑒−2𝑥𝑥2 +

2[pic 13]

1        2

(−2𝑒−2𝑥𝑥 − 𝑒−2𝑥)] 4        0[pic 14]

Calculo limites:

lim → 0 + (− 1 𝑒−2𝑥𝑥2 + 1 (−2𝑒−2𝑥𝑥 − 𝑒−2𝑥))[pic 15][pic 16]

𝑥        2        4

= − 1 𝑒−2×0 × 02 + 1 (−2𝑒−2×0 × 0 − 𝑒−2×0) = − 1[pic 17][pic 18][pic 19]

2        4        4

lim → 2 + (− 1 𝑒−2𝑥𝑥2 + 1 (−2𝑒−2𝑥𝑥 − 𝑒−2𝑥))[pic 20][pic 21]

𝑥        2        4

= − 1 𝑒−2×2 × 22 + 1 (−2𝑒−2×2 × 2 − 𝑒−2×2) = − 13[pic 22][pic 23][pic 24]

2        4        4𝑒4

13        1        13

= 4 (− 4𝑒4 + 4) = − 𝑒4 + 1[pic 25][pic 26][pic 27]

...