Técnicas de Integración: Aprende a Integrar con Sustitución, Partes y Fracciones Parciales

Question 1

Calcula la integral de (x^3 + 2x^2 - 1)e^(x^2 + x) dx.

Accepted Answer

((x^3 + 2x^2 - 1) / 2)e^(x^2 + x) + C

Answer

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

Answer

((x^3 + x^2 - 1) / 2)e^(x^2 + x) + C

Answer

(x^4 + x^3 - x)e^(x^2 + x) + C

Question 2

Calcula la integral de ∫(x^2 + 1)/(x^3 + x) dx.

Accepted Answer

ln|x^3 + x| + C

Answer

-ln|x^3 + x| + C

Answer

1/(x^3 + x) + C

Answer

-1/(x^3 + x) + C

Question 3

Calcula la integral de ∫(x^3 - 2x)/(x^2 - 1) dx.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 4

Calcula la integral de ∫(x^2 + 1)/(x^3 - x) dx.

Accepted Answer

-1/2 ln|x^3 - x| + 1/x + C

Answer

1/2 ln|x^3 - x| + 1/x + C

Answer

-1/2 ln|x^3 - x| - 1/x + C

Answer

1/2 ln|x^3 - x| - 1/x + C

Question 5

Calcular la integral indefinida de la función f(x) = x^2 + 3x - 2 respecto de x:

Accepted Answer

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

Answer

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

Answer

x + 3/2 - 2 + C

Answer

3x^2 + (9x)/2 - 6x - C

Question 6

Resolver la integral indefinida de f(x) = x / (x^2 + 1) mediante integración por fracciones parciales:

Accepted Answer

1/2 ln(|x^2 + 1|) + C

Answer

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

Answer

x / (x^2 + 1) + C

Answer

-1/2 ln(|x^2 + 1|) + C

Question 7

Integrar la función f(x) = x*e^(2x) por partes:

Accepted Answer

(x/2)*e^(2x) - (1/4)*e^(2x) + C

Answer

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

Answer

e^(2x) + C

Answer

(x/2)*e^(2x) + (1/4)*e^(2x) + C

Question 8

Evaluar la integral indefinida de f(x) = cos(x)*sin(x):

Accepted Answer

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

Answer

-sin^2(x) + C

Answer

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

Answer

cos^2(x) + C

Question 9

Integrar la función f(x) = (x^2 - 2x + 3) / (x - 1) por fracciones parciales:

Accepted Answer

x^2 - x + 4 + ln(|x - 1|) + C

Answer

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

Answer

x^2 + 4 + C

Answer

x^2 - x + 4 - ln(|x - 1|) + C

Question 10

Calcular la integral de (x^2 + 2x - 1):

Accepted Answer

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

Answer

x^2/2 + x - 1 + C

Answer

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

Answer

2x^3 + x^2 - x + C

Question 11

Calcular la integral de (sin x + cos x):

Accepted Answer

-cos x + sin x + C

Answer

-sin x - cos x + C

Answer

cos x - sin x + C

Answer

sin x + cos x - C

Question 12

Calcular la integral de (e^x + 1/x):

Accepted Answer

e^x + ln |x| + C

Answer

e^x - ln |x| + C

Answer

-e^x + ln |x| + C

Answer

2e^x + ln |x| + C

Question 13

Calcular la integral de (x^2 / (x^2 + 1)):

Accepted Answer

x - arctan x + C

Answer

-x + arctan x + C

Answer

x - arctan (1/x) + C

Answer

x + arctan x + C

Question 14

Calcular la integral de (e^x sin x):

Accepted Answer

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

Answer

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

Answer

e^x (sin x + cos x) + C

Question 15

Calcular la integral de (x^2 / (x^3 + 1)):

Accepted Answer

(1/3) ln |x^3 + 1| + C

Answer

(1/3) ln |x| + C

Answer

(1/3) arctan (x^3 + 1) + C

Answer

-ln |x| + C

Question 16

Calcular la integral de (1 / sqrt(x^2 + 1)):

Accepted Answer

ln |x + sqrt(x^2 + 1)| + C

Answer

arctan (sqrt(x^2 + 1)) + C

Answer

ln |x| + C

Answer

-ln |x + sqrt(x^2 + 1)| + C

Question 17

Calcular la integral de (cos^2 x):

Accepted Answer

(1/2) x + (1/4) sin 2x + C

Answer

-x + sin 2x + C

Answer

(1/2) x - (1/4) sin 2x + C

Answer

x + sin 2x + C

Question 18

Calcular la integral de (x e^x):

Accepted Answer

x e^x - e^x + C

Answer

x e^x + e^x + C

Answer

-x e^x + e^x + C

Answer

x e^x - 2e^x + C