Help needed with Calculus

Question

Find the average value of  over the interval [0, 4].

Options:

1)	1.61
2)	0.40
3)	4.83
4)	-0.20

Answer 1

f(x) = 1/(x+1) on the interval [0,4]

The average value of a function is

1/(b-a) * ∫ f(x)dx <-- The definite integral goes from a lower value a to an upper value b

1/(4-0) * ∫ 1/(x+1)dx

= 1/4 * ∫ 1/(x+1)dx

We use u-substitution

u = x+1

du = dx

∫ 1/(x+1)dx = ∫ 1/u du

= ln(u)

Next re-substitute back in u = x+1

∫ 1/(x+1)dx = ln(x+1)

Finally plug in x = 4 and x = 0 to find the definite integral

ln(4+1) - ln(0+1) = ln(5)

Therefore the average value is 1/4*ln(5) = 0.402