What is prmitive of function 1/(cosx)^4?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the primitive of the given function, hence, you need to evaluate the indefinite integral of the function, such that:

`int 1/(cos^4 x)dx`

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

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

Replacing ` 1 + tan^2 x` for `1/(cos^2 x)` yields:

`int 1/(cos^4 x)dx = int (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 variable yields:

`int (1 + tan^2 x)*1/(cos^2 x) dx = int (1 + t^2)*dt`

Using the property of linearity of integral, yields:

`int (1 + t^2)*dt = int dt + int t^2 dt`

`int (1 + t^2)*dt = t + t^3/3 + c`

Replacing back `tan x` for `t` yields:

`int 1/(cos^4 x)dx = tan x + (tan^3 x)/3 + c`

Hence, evaluating the requested primitive of the given function, yields `int 1/(cos^4 x)dx = tan x + (tan^3 x)/3 + c.`