To determine

To find:

The average value of the given function on the given interval.

Answer

25π

Explanation

1) Concept:

Use the formula for the average value of h on the interval [a, b].

2) Formula:

have=1b-a∫abh(x)dx

3) Given:

hx=cos4⁡xsin⁡x,     [0, π]

4) Calculation:

Here, hx=cos4⁡xsin⁡x,  a=0 and b=π

Substituting all the above in the formula,

have=1π-0∫0πcos4⁡xsin⁡xdx

Simplify.

have=1π∫0πcos4⁡xsin⁡xdx

Here, we need to use the substitution method.

Let u=cos⁡x, then du=-sin⁡xdx, so sin⁡xdx=-du

Since this is a definite integral, the limits of the integration also change.

When x=0, u=1 and x=π, u=-1

Therefore, the integration becomes

have=1π∫0πcos4⁡xsin⁡xdx

=1π∫1-1u4-du

Using the property of reversing order of definite integral,

= 1π∫-11u4du

Since u4 is an even function, use the property of definite integral for even functions.

=2π∫01u4du

Integrating,

=2πu5501

Simplifying,

=25πu501

=25π15-05

=25π

Therefore,

have=25π

Conclusion:

The average value of hx=cos4⁡xsin⁡x on the interval [0, π] is 25π