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documento de derivasdasd, Schemes and Mind Maps of Law

Escuela Tecnica de Electricidad Law

Prof. Romo Medina

parctica derviadas e integrales

Typology: Schemes and Mind Maps

2024/2025

Uploaded on 03/28/2025

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1

MECÁNICA NEWTONIANA

2021 A

HOJA DE TRABAJO 3

CÁLCULO INTEGRAL

Calcule las siguientes integrales indefinidas y compruebe el resultado por derivación:

1. ∫√𝑥2

3 𝑑𝑥. R: 3

5𝑥5

3+𝐶

2. ∫(2sin𝑥+𝑒𝑥) 𝑑𝑥. R: −2cos𝑥+𝑒𝑥+𝐶

3. ∫(5𝑥√𝑥 − 3cos𝑥) 𝑑𝑥. R: 2𝑥5

2−3𝑠𝑒𝑛𝑥+𝐶

4. ∫4𝑥

√16−𝑥2 𝑑𝑥. R: −4(16−𝑥2)1

2+𝐶

5. ∫√𝑥5

3𝑥−4/3(𝑥3− 1) 𝑑𝑥.

Use el cambio de variable 𝒖=𝒈(𝒙) para calcular:

6. ∫cos3𝑥 sin𝑥 𝑑𝑥, 𝑢=cos𝑥. R: −cos4𝑥

4+𝐶

7. ∫𝑥2 𝑒𝑥3 𝑑𝑥, 𝑢=𝑥3. R:1

3𝑒𝑥3+𝐶

8. ∫𝑥

√𝑥2+3 𝑑𝑥, 𝑢=𝑥2+3. R: (𝑥2+3)1

2+𝐶

9. ∫sec4𝑥 tan3𝑥 𝑑𝑥, 𝑢=tan𝑥 R: 𝑡𝑎𝑛4𝑥

4+tan6𝑥

6+𝐶

10. ∫𝑥3

√25−𝑥4 𝑑𝑥, 𝑢=25−𝑥4.

Calcule mediante integración por partes:

11. ∫ln(𝑎𝑥) 𝑑𝑥. R: 𝑥 𝑙𝑛𝑎𝑥−𝑥+𝐶

12. ∫𝑥 𝑒𝑎𝑥 𝑑𝑥. R: 𝑥

𝑎𝑒𝑎𝑥−1

𝑎2𝑒𝑎𝑥+𝐶

13. ∫𝑥 cos2𝑥 𝑑𝑥. R: 1

4𝑥2+1

4𝑥sen2𝑥+1

8cos2𝑥+𝐶

14. ∫𝑥√𝑥 + 1 𝑑𝑥. R: 2

3𝑥(𝑥+1)3

2−4

15(𝑥+1)5

2+𝐶

Calcule las siguientes integrales definidas:

15. ∫2

15−𝑥

𝑥3 𝑑𝑥. R: 1.375

16. ∫1

0(2𝑡+1)4 𝑑𝑡. R: 24.2

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MECÁNICA NEWTONIANA

2021 A

HOJA DE TRABAJO 3

CÁLCULO INTEGRAL

Calcule las siguientes integrales indefinidas y compruebe el resultado por derivación:

2

3

𝑑𝑥. R:

3

5

3

𝐶

(2sin𝑥 + 𝑒

𝑥

) 𝑑𝑥. R: − 2 cos 𝑥 + 𝑒

𝑥

𝑥 − 3cos𝑥) 𝑑𝑥. R: 2 𝑥

5

2 − 3 𝑠𝑒𝑛𝑥 + 𝐶

4 𝑥

√ 16 −𝑥

2

𝑑𝑥. R: − 4 ( 16 − 𝑥

2

1

2

𝐶

5

3

− 4 / 3

3

Use el cambio de variable 𝒖 = 𝒈(𝒙) para calcular:

∫ cos

3

𝑥 sin𝑥 𝑑𝑥, 𝑢 = cos𝑥. R: −

cos

4

𝑥

4

2

𝑥

3

. R:

1

3

𝑥

3

𝑥

√𝑥

2

3

2

+ 3. R:

2

1

2

𝐶

sec

4

𝑥 tan

3

𝑥 𝑑𝑥, 𝑢 = tan𝑥 R:

𝑡𝑎𝑛

4

𝑥

4

tan

6

𝑥

6

𝑥

3

√ 25 −𝑥

4

Calcule mediante integración por partes:

∫ ln(𝑎𝑥) 𝑑𝑥. R: 𝑥 𝑙𝑛 𝑎𝑥 − 𝑥 + 𝐶

𝑎𝑥

𝑑𝑥. R:

𝑥

𝑎

𝑎𝑥

1

𝑎

2

𝑎𝑥

∫ 𝑥 cos

2

𝑥 𝑑𝑥. R:

1

4

2

1

4

𝑥 sen 2 𝑥 +

1

8

cos 2 𝑥 + 𝐶

14. ∫ 𝑥√𝑥 + 1 𝑑𝑥. R:

2

3

2

−

4

15

5

2

𝐶

Calcule las siguientes integrales definidas:

2

1

5 −𝑥

𝑥

3

𝑑𝑥. R: 1. 375

1

0

4

𝑑𝑡. R: 24. 2

𝑎

0

2

𝑑𝑥, donde 𝑎 es constante. R:

1

6

2

𝑣

0

𝑑𝑣

𝑔−

𝑘

𝑚

𝑣

, donde 𝑔, 𝑘, 𝑚 son constantes positivas. R:

𝑚

𝑘

𝑔−

𝑘

𝑚

𝑣

0

𝑔−

𝑘

𝑚

𝑣

0

𝑑𝑣

𝑔−

𝑘

𝑚

𝑣

2

, donde 𝑔, 𝑘, 𝑚 son constantes positivas. R:

1

2

𝑚

𝑔𝑘

ln |

√

𝑘

𝑚

𝑣+

√

𝑔

√

𝑘

𝑚

𝑣− √

𝑔

Ejercicios adicionales:

Calcule ∫

𝜋

0

𝑥

cos𝑒

𝑥

𝑑𝑥. R: − 1. 75

Calcule ∫

𝜋

0

sin2𝑥cos2𝑥 𝑑𝑥. R: 0

Determine la integral ∫

5

1 +𝑒

𝑥

𝑑𝑥, sabiendo que si 𝑥 = 1 , el valor de la integral es 8.

R: 5 𝑙𝑛 ( 1 −

1

1 +𝑒

𝑥

Realice un esquema de la región acotada superiormente por la curva 𝑦 = √

𝑥 e inferiormente por la curva

𝑥

4

, y determine su área. R:

32

3

u

2

Una curva que pasa por el punto ( 4 ,

1

3

) unidades, posee una pendiente variable de

𝑑𝑦

𝑑𝑥

1

2 √𝑥( 1 +√𝑥)

2

a) Halle la ecuación de la curva. R:

1

1 + √

𝑥

b) Represente gráficamente la curva.

c) Calcule el área bajo la curva en el intervalo [ 0 , 1 ] unidades. R: 0. 614 u

2

Calcule la longitud de la curva 𝑦 = 𝑥

2

, para 0 ≤ 𝑥 ≤ 20. Utilice una integral de tabla.

R: 401. 22 u

documento de derivasdasd, Schemes and Mind Maps of Law

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Partial preview of the text

Download documento de derivasdasd and more Schemes and Mind Maps Law in PDF only on Docsity!

MECÁNICA NEWTONIANA

2021 A

HOJA DE TRABAJO 3

CÁLCULO INTEGRAL

𝑑𝑥. R:

𝑑𝑥. R: − 4 ( 16 − 𝑥

. R:

+ 3. R:

𝑑𝑥. R:

𝑥 𝑑𝑥. R:

14. ∫ 𝑥√𝑥 + 1 𝑑𝑥. R:

𝑑𝑥. R: 1. 375

𝑑𝑡. R: 24. 2

𝑑𝑥. R: − 1. 75

R: 5 𝑙𝑛 ( 1 −