What is antiderivative of y=tg x?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the anti-derivative of the given function `y = tan x` , hence, you need to evaluate the following indefinite integral, such that:

`int ydx = int tan x dx`

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

`tan x = sin x/cos x`

Replacing `sin x/cos x` for `tan x` yields:

`int tan x dx = intsin x/cos x dx`

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

`cos x = t => -sin xdx = dt => sin x dx = -dt`

Changing the variable yields:

`intsin x/cos x dx = int (-dt)/t`

Taking out the constant yields:

`int (-dt)/t = - int (dt)/t = -ln|t| + c`

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

`int tan x dx = -ln|cos x| + c `

Using theĀ  power property of logarithms yields:

`int tan x dx = ln|cos x|^(-1) + c `

`int tan x dx = ln (1/|cos x|) + c`

Hence, evaluating the anti-derivative of the given function, using substitution method and the logarithmic properties, yields `int tan x dx = ln (1/|cos x|) + c.`

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