Question

How do you find the antiderivative of `tan(6x)dx`?

Answer 1

`I=1/6ln|sec6x|+c`

Explanation:

We know that
`inttanXdX=ln|secX|+c`
Here,
`I=inttan6xdx`
Let, `6x=u=>x=u/6=>dx=1/6du`
So,
`I=inttanu1/6du`
`=1/6inttanudu`
`=1/6ln|secu|+c ,where, u=6x`
`=1/6ln|sec6x|+c`

Answer 2

`(ln|sec(x)|)/6+C`

Explanation:

We need to find the anti-derivative of `tan(6x)`, and that will be its integral, i.e. `inttan(6x) \ dx`.

So, we gonna use u-substitution here.

Let `u=6x,:.du=6 \ dx,dx=(du)/6`

Then,

`inttan(6x) \ dx`

`=inttanu \ (du)/6`

Taking out the constant, we get,

`=1/6inttanu \ du`

Now, notice that `inttanu \ du=ln|sec(x)|+C`.

And so, we get,

`=1/6*ln|sec(x)|+C`

`=(ln|sec(x)|)/6+C`