Calculate `int_0^(pi/4) 1/(cosx)^4 ` ?

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You need to evaluate the definite integral, such that:

`int_0^(pi/4) 1/(cos^4 x)dx = int_0^(pi/4) 1/(cos^2 x)*1/(cos^2 x)dx `

You need to use the following trigonometric identity, such that:

`1/(cos^2 x) = 1 + tan^2 x`

You need to replace `1 + tan^2 x` for `1/(cos^2 x)` such that:

`int_0^(pi/4) (1 + tan^2 x)*1/(cos^2 x)dx `

You need to come up with the following substitution, such that:

`tan x = t => 1/(cos^2 x) dx = dt`

Changing the limits of integration yields:

`x = 0 => t = tan 0 = 0`

`x = pi/4 => t = tan (pi/4) = 1`

Replacing the variable, yields:

`int_0^1 (1 + t^2)*dt`

Using the property of linearity of integral yields:

`int_0^1 dt + int_0^1 (t^2)*dt = (t + t^3/3)|_0^1`

You need to use the fundamental formula of calculus, yields:

`int_0^1 dt + int_0^1 (t^2)*dt = (1 + 1/3 - 0 - 0/3)`

`int_0^1 dt + int_0^1 (t^2)*dt = 4/3`

Hence, evaluating the definite integral, yields `int_0^(pi/4) 1/(cos^4 x)dx = 4/3.`

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