What is the average value of the function $f (x) = sec x tan x$ $f(x) = sec x tan x$ on the interval $[0, \frac{π}{4}]$ $[0,pi/4]$ ?

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What is the average value of the function $f (x) = sec x tan x$ $f(x) = sec x tan x$ on the interval $[0, \frac{π}{4}]$ $[0,pi/4]$ ?

1 Answer

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Answer 1 · 2016-03-23T17:13:32

It is $\frac{4 (\sqrt{2} - 1)}{π}$ $(4(sqrt2-1))/pi$

Explanation:

The average value of a function $f$ $f$ on an interval $[a, b]$ $[a,b]$ is

$\frac{1}{b - a} \int_{a}^{b} f (x) d x$ $1/(b-a) int_a^b f(x) dx$

So the value we seek is

$\frac{1}{\frac{π}{4} - 0} \int_{0}^{\frac{π}{4}} sec x tan x d x$ $1/(pi/4-0) int_0^(pi/4) secxtanx dx$

$= \frac{4}{π} {[sec x]}_{0}^{\frac{π}{4}}$ $= 4/pi [secx]_0^(pi/4)$

$= \frac{4}{π} [sec (\frac{π}{4}) - sec (0)]$ $= 4/pi [sec(pi/4)-sec(0)]$

$= \frac{4}{π} [\sqrt{2} - 1]$ $= 4/pi [sqrt2-1]$

$= \frac{4 (\sqrt{2} - 1)}{π}$ $= (4(sqrt2-1))/pi$

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What is the average value of the function $f (x) = sec x tan x$ $f(x) = sec x tan x$ on the interval $[0, \frac{π}{4}]$ $[0,pi/4]$ ?

1 Answer

Explanation:

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What is the average value of the function $f (x) = sec x tan x$ $f(x) = sec x tan x$ on the interval $[0, \frac{π}{4}]$ $[0,pi/4]$ ?