Question

How do you find the antiderivative of `int sec^2xtanx dx`?

Answer 1

`intsec^2xtanxdx=1/2sec^2x+c`

Explanation:

Method 1

now `d/(dx)(secx)=secxtanx`

so`intsec^2xtanxdx=1/2sec^2x+c_1`

since`d/(dx)(sec^2x)=2(secx)xxsecxtanx`

by the chain rule

Method 2

`d/(dx)(tan^2x)=2(tanx)xxsec^2x`

by the chain rule

so`intsec^2xtanxdx=1/2tan^2x+c_2`

the 2 results can be shown to be equivalent by use of

`1+tan^2x=sec^2x`