Reglas de Integración: Aprende a Integrar Funciones Elementales

Question 1

Calcular la integral indefinida de ∫(x² + 3x - 2) dx

Accepted Answer

x³/3 + 3x²/2 - 2x + C

Answer

x³/2 + 3x²/4 - 2x + C

Answer

x⁴/3 + 3x³/2 - 2x + C

Answer

x⁴/4 + 3x³/3 - 2x² + C

Question 2

Calcular ∫(e^x + 1/x) dx

Accepted Answer

e^x + ln(|x|) + C

Answer

e^x + ln(x) + C

Answer

-e^x + ln(|x|) + C

Answer

e^x - ln(|x|) + C

Question 3

Evaluar ∫(1/(x-1)) dx

Accepted Answer

ln(|x-1|) + C

Answer

ln(x-1) + C

Answer

-ln(x-1) + C

Answer

-ln(|x-1|) + C

Question 4

Calcular la integral definida de ∫[0,1] (x² + 2x) dx

Accepted Answer

1,33

Answer

0,33

Answer

2,33

Answer

0,67

Question 5

Calcular la integral por partes de ∫(x * e^x) dx

Accepted Answer

x * e^x - e^x + C

Answer

-x * e^x + e^x + C

Answer

x * e^x - 2e^x + C

Answer

x * e^x + e^x + C

Question 6

Evaluar ∫(1 / (x² + 1)) dx

Accepted Answer

arctan(x) + C

Answer

-arctan(x) + C

Answer

-arctan(x) + C

Question 7

¿Cuál es el enunciado de la regla de integración por fracciones parciales?

Accepted Answer

Descomponer la fracción en fracciones más simples con denominadores lineales o cuadráticos.

Answer

Multiplicar la fracción por su conjugado.

Answer

Integrar la fracción directamente.

Answer

Factorizar el denominador.

Question 8

Calcula la integral indefinida ∫(1/(x - 1)(x + 2)) dx usando fracciones parciales.

Accepted Answer

ln|x - 1| - ln|x + 2| + C

Answer

(x - 1)/(x + 2) + C

Answer

1/(x - 1) - 1/(x + 2) + C

Answer

1/(x - 1) + 1/(x + 2) + C

Question 9

Evalúa la convergencia o divergencia de la integral impropia ∫(x/(x^2 + 1)) dx.

Accepted Answer

Convergente

Answer

Divergente

Answer

Absolutamente convergente

Answer

Semiconvergente

Question 10

La regla de potencia de integración establece que, para cualquier constante real a y cualquier entero positivo n, la integral indefinida de ax^n es:

Accepted Answer

∫ ax^n dx = (a/n+1)x^(n+1) + C

Answer

∫ ax^n dx = (a/n)x^n + C

Answer

∫ ax^n dx = (an)x^(n+1) + C

Answer

∫ ax^n dx = ax^(n+1) + C

Question 11

¿Cuál es la regla para calcular la integral indefinida de funciones de la forma ∫ 1/x dx?

Accepted Answer

∫ 1/x dx = ln|x| + C

Answer

∫ 1/x dx = x + C

Answer

∫ 1/x dx = 1 + C

Answer

∫ 1/x dx = -1/x + C

Question 12

Integra la función f(x) = x^3 + 2x:

Accepted Answer

(x^4)/4 + x^2 + C

Answer

(x^3)/4 + C

Answer

x^3 + x^2 + C

Answer

x^2 + 2x + C

Question 13

Calcula el valor de la integral definida ∫[0,1] x^2 dx:

Accepted Answer

1/3

Answer

1

Answer

0

Answer

2

Question 14

¿Cuál es la integral indefinida de f(x) = e^x?

Accepted Answer

e^x + C

Answer

e^-x + C

Answer

x e^x + C

Answer

1/e^x + C

Question 15

La regla de sustitución para integrar funciones de la forma ∫ f(g(x)) g'(x) dx es:

Accepted Answer

u = g(x), du/dx = g'(x)

Answer

u = g'(x), du/dx = f(g(x))

Answer

u = f(g(x)), du/dx = f'(g(x))

Answer

u = f(x), du/dx = g(x)

Question 16

¿Cuál de las siguientes reglas de integración permite integrar la función f(x) = x^2?

Accepted Answer

∫x^n dx = (x^(n+1))/(n+1) + C

Answer

∫e^x dx = e^x + C

Answer

∫cos(x) dx = sin(x) + C

Answer

∫sin(x) dx = -cos(x) + C

Question 17

Calcula la integral indefinida de ∫(3x - 2) dx.

Accepted Answer

(3/2)x^2 - 2x + C

Answer

(3/2)x^2 + 2x + C

Answer

(6/2)x^2 - 2x + C

Answer

(6/2)x^2 + 2x + C

Question 18

Integra la función f(x) = e^(2x).

Accepted Answer

(1/2)e^(2x) + C

Answer

2e^(2x) + C

Answer

(1/2)e^(x) + C

Answer

2e^(x) + C

Question 19

Integra la función f(x) = 1/(x+1).

Accepted Answer

ln|x+1| + C

Answer

ln|x-1| + C

Answer

-ln|x+1| + C

Question 20

Si $f'(x) = cos(x)$, ¿cuál de las siguientes opciones podría ser $f(x)$?

Accepted Answer

$sin(x) + C$

Answer

$-sin(x) + C$

Answer

$tan(x) + C$

Answer

$sec^2(x) + C$

Question 21

Encuentra la integral indefinida de $\frac{1}{x-2}$ dx.

Accepted Answer

$ln|x-2| + C$

Answer

$arctan(x-2) + C$

Answer

$x + ln|x-2| + C$

Answer

$\frac{1}{(x-2)^2} + C$

Question 22

¿Cuál es la integral de $e^{2x}$ dx?

Accepted Answer

$\frac{1}{2}e^{2x} + C$

Answer

$e^{2x} + C$

Answer

$2e^{2x} + C$

Answer

$\frac{1}{4}e^{2x} + C$

Question 23

Utiliza la integración por partes para encontrar la integral indefinida de $x \cdot sen(x)$ dx.

Accepted Answer

$-x \cdot cos(x) + sen(x) + C$

Answer

$\frac{1}{2}x^2sen(x) - sen(x) + C$

Answer

$\frac{1}{2}x^2sen(x) + C$

Answer

$x \cdot cos(x) + sen(x) + C$

Question 24

Utiliza la sustitución trigonométrica para encontrar la integral indefinida de $\sqrt{1-x^2}$ dx.

Accepted Answer

$arcsin(x) + \frac{x}{\sqrt{1-x^2}} + C$

Answer

$arcsin(x) + \frac{x}{2\sqrt{1-x^2}} + C$

Answer

$2arcsin(x) + \frac{x}{\sqrt{1-x^2}} + C$

Question 25

¿Cuál es la integral indefinida de la función f(x) = 2x² + 3x - 1?

Accepted Answer

2/3 x³ + 3/2 x² - x + C

Answer

x³ + x² - C

Answer

4x + 6 + C

Answer

3x³ + 2x² - 1 + C

Question 26

¿Cuál es la integral indefinida de  j(x) = sin(2x)?

Accepted Answer

-1/2 cos(2x) + C

Answer

1/2 cos(2x) + C

Answer

-sin(2x) + C

Answer

sin(2x) + C

Question 27

Cuál es la integral indefinida de la función g(x) = 1/(1 + x²)?

Accepted Answer

arctan(x) + C

Answer

-1/(1 + x²)² + C

Answer

2x/(1 + x²)² + C

Answer

ln(1 + x²) + C

Question 28

Calcula la integral indefinida de la función m(x) = 1/x.

Accepted Answer

∫m(x)dx = ln|x| + C

Answer

∫m(x)dx = x + C

Answer

∫m(x)dx = 1/x² + C

Answer

∫m(x)dx = -1/x² + C

Question 29

Aplica la regla de la cadena para calcular la integral de la función p(x) = (2x + 1)³.

Accepted Answer

∫p(x)dx = (2x + 1)⁴/8 + C

Answer

∫p(x)dx = (2x + 1)⁴ + C

Answer

∫p(x)dx = 3(2x + 1)² + C

Answer

∫p(x)dx = 6(2x + 1)² + C

Question 30

Calcula la integral definida de la función q(x) = x²  entre los límites x = 0 y x = 2.

Accepted Answer

∫₀² q(x)dx = 8/3

Answer

∫₀² q(x)dx = 16/3

Answer

∫₀² q(x)dx = 4

Answer

∫₀² q(x)dx = 2

Question 31

Determina la integral indefinida de la función s(x) = e^x.

Accepted Answer

∫s(x)dx = e^x + C

Answer

∫s(x)dx = 1/e^x + C

Answer

∫s(x)dx = e^(x+1) + C

Answer

∫s(x)dx = x*e^x + C

Question 32

Determina la integral indefinida de la función ( t(x) = sin(x). )

Accepted Answer

∫t(x)dx = -cos(x) + C

Answer

∫t(x)dx = cos(x) + C

Answer

∫t(x)dx = -sin(x) + C

Answer

∫t(x)dx = sin(x) + C

Question 33

Calcula la integral indefinida de la función ( f(x) = 3x² + 2x - 1. )

Accepted Answer

x³ + x² - x + C

Answer

x³ + x² - 1 + C

Answer

9x² + 2 + C

Answer

6x + 2 + C

Question 34

Encuentra la integral indefinida de la función ( f(x) = e^x + rac{1}{x}. )

Accepted Answer

e^x + ln|x| + C

Answer

e^x - ln|x| + C

Answer

e^x - rac{1}{x²} + C

Answer

e^x + rac{1}{x²} + C

Question 35

Calcula la integral indefinida de la función ( f(x) = sin(x) + cos(x). )

Accepted Answer

-cos(x) + sin(x) + C

Answer

cos(x) - sin(x) + C

Answer

cos(x) + sin(x) + C

Answer

-cos(x) - sin(x) + C

Question 36

Calcula la integral definida de la función ( f(x) = x² ) de ( 0 ) a ( 2 ).

Accepted Answer

8/3

Answer

4

Answer

2

Question 37

Encuentra la integral indefinida de la función ( f(x) = (2x + 1)² ).

Accepted Answer

(2/3)x³ + x² + x + C

Answer

(2/3)x³ + 2x² + x + C

Answer

4x³ + x² + x + C

Answer

4x³ + 2x² + x + C

Question 38

Calcula la integral indefinida de la función (f(x) = x * cos(x²) ).

Accepted Answer

(1/2)sin(x²) + C

Answer

- sin(x²) + C

Answer

sin(x²) + C

Answer

(1/2)cos(x²) + C

Question 39

Calcula la integral indefinida de la función ( f(x) = rac{1}{x² + 1} ).

Accepted Answer

arctan(x) + C

Answer

rac{1}{x} + C

Answer

- rac{1}{x² + 1} + C

Answer

ln(x² + 1) + C

Question 40

Calcula la integral definida de la función ( f(x) = x * e^x ) de ( 0 ) a ( 1 ).

Accepted Answer

e - 1

Answer

e + 1

Answer

1

Answer

e

Question 41

Encuentra la integral indefinida de la función ( f(x) = ln(x). )

Accepted Answer

x * ln(x) - x + C

Answer

ln(x) + C

Answer

rac{1}{x} + C

Answer

x * ln(x) + C

Question 42

Calcula la integral definida de la función f(x) = 1/x de 1 a e.

Accepted Answer

1

Answer

ln(e)

Answer

0

Answer

e