What is the definite integral of (x - 4)^3 with respect to x, evaluated from 1 to 4?

To find the definite integral of ( (x - 4)^3 ) from 1 to 4, you first need to compute the indefinite integral of the function.

1. Finding the Indefinite Integral:

The function to integrate is ( (x - 4)^3 ). To find the indefinite integral, you can use the power rule of integration:

[

\int (x - 4)^3 , dx = \frac{(x - 4)^4}{4} + C

]

Here, ( C ) is the constant of integration.

2. Evaluating the Integral from 1 to 4:

Now, substitute the limits of integration (1 and 4) into the indefinite integral:

[

\left[ \frac{(x - 4)^4}{4} \right]_{1}^{4}

]

First, evaluate at the upper limit (x = 4):

[

\frac{(4 - 4)^4}{4} = \frac{0^4}{4} = 0

]

Next, evaluate at the lower