What is antiderivative y=((tgx)^2+(tgx)^4)?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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

`int f(x)dx = int (tan^2 x + tan^4 x)dx`

You need to factor out `tan^2 x` , such that:

`int f(x)dx = int tan^2 x(1 + tan^2 x)dx`

You should use the following trigonometric identity, such that:

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

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

`int f(x)dx = int tan^2 x*(dx)/(cos^2 x)`

You should come up with the following substitution, such that:

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

Replacing the variable yields:

`int t^2 dt = t^3/3 + c`

Replacing back `tan x` for t yields:

`int (tan^2 x + tan^4 x)dx= (tan^3 x)/3 + c`

Hence, evaluating the anti-derivative of the given function, yields `int (tan^2 x + tan^4 x)dx = (tan^3 x)/3 + c`