Question

What is the integral of this function?

`int ((e^tan(3x))/(cos^2(3x)))dx`

I'm confused about this problem

Answer 1

Is an inmediate integral. See below

Explanation:

You know that `d(tanx)/dx=sec^2x=1/cos^2x`

You also know that `inte^(u)u´du=e^(u)+C`

Appliying this, we have

`inte^(tan(3x))/cos^2(3x)dx=1/3e^(tan3x)+C`

Answer 2

=`e^tan(3x)+C`

Explanation:

First
`color(blue)(1/cos^2(3x)=sec^2(3x)`

`inte^tan(3x)/cos^2(3x)dx=inte^tan(3x)sec^2(3x)dx`

`color(blue)(d/dxtan(3x)=sec^2(3x)dx`

`color(blue)(inte^udu=e^u+C` `" "` `color(green)(tan(3x)=u" " "in this case"`

Thus

`inte^tan(3x)sec^2(3x)`=`e^tan(3x)+C`