# What is the average value of the function # f(x) = 4x ln (2x) # on the interval #[1,4]#?

So here, the average value is equal to

Let's first find this integral without the bounds.

So:

Simplifying and integrating:

This is without the constant go integration because we will use this to evaluate the integral. Returning to the average value, we see that

Now evaluating in full:

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To find the average value of the function ( f(x) = 4x \ln(2x) ) on the interval ([1,4]), we use the formula for the average value of a function over an interval:

[ \text{Average value} = \frac{1}{b - a} \int_a^b f(x) , dx ]

Where ( a ) and ( b ) are the endpoints of the interval.

So, for the function ( f(x) = 4x \ln(2x) ) on the interval ([1,4]), we have:

[ a = 1 ] [ b = 4 ]

Now, we evaluate the integral:

[ \int_1^4 4x \ln(2x) , dx ]

Then we divide the result by the length of the interval:

[ \frac{1}{4 - 1} \int_1^4 4x \ln(2x) , dx ]

[ = \frac{1}{3} \int_1^4 4x \ln(2x) , dx ]

This integral may require integration by parts. After calculating the integral, we obtain a numerical value. Finally, we divide this numerical value by 3 to get the average value of the function over the interval ([1,4]).

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