What is the result of the integral of the function f(x) = 6x from 1 to 3?

To find the result of the integral of the function f(x) = 6x from 1 to 3, we start by calculating the definite integral:

[

\int_{1}^{3} 6x , dx

]

First, we find the antiderivative of 6x. The antiderivative is given by:

[

\int 6x , dx = 3x^2 + C

]

where C is the constant of integration (which we will not need for definite integrals). Next, we evaluate this antiderivative at the upper limit (3) and the lower limit (1):

[

\text{Evaluate at } 3: \quad 3(3^2) = 3(9) = 27

]

[

\text{Evaluate at } 1: \quad 3(1^2) = 3(1) = 3

]

Now, subtract the value at the lower limit from the value at the upper limit:

[

27 - 3 = 24

]

Thus, the result of the integral from 1 to 3 is 24, which matches the value indicated in the first