How do you evaluate the definite integral int (2x^3)dx from [1,3]?

1 Answer
Dec 11, 2016

int_1^3 2x^3dx = 40

Explanation:

First, we will consider the integral as I with the limits:

I = int_1^3 2x^3dx

We will take out the constant as it is in multiplication with the variable:

I = 2 int_1^3 x^3dx

We know the power rule of integration:

I = int x^ndx = (x^(n+1))/(n+1)

Applying power rule on the integral:

I = 2 [(x^(3+1))/(3+1)]_1^3

I = 2 [(x^4)/4]_1^3

Now we can put the limits of the integration, we know the rule of limits of integration:

Upper Limit-Lower Limit

Hence:

I = 2 [((3^4)/4)-(1^4/4)]

We know, 3^4=81 and 1^4=1

I = 2 [((81)/4)-(1/4)]

As the base is same, we can directly subtract 81 and 1

I = 2 [((81-1)/4)]

I = 2 [80/4]

We can divide 80 by 4

I = 2.20

I=40

Hence:

int_1^3 2x^3dx = 40