Question

What is the antiderivative of `1/cosx(2+sinx)`?

Answer 1

`int frac{sinx+2}{cosx}dx`
`=ln|secx|+2ln|secx+tanx|+C`

Explanation:

`int frac{sinx+2}{cosx}dx`

`int frac{sinx}{cosx}+frac{2}{cosx}dx`

`=int (tanx)dx + 2int(secx)dx`

`=ln|secx|+2ln|secx+tanx|+C`

Use logarithm rules:
`=ln|secx|+ln(secx+tanx)^2+C`

`=ln|(secx)(secx + tanx)^2|+C`