Question

What is `int(sinx)/(cos^2 x) dx`?

Answer 1

`secx+C`

Explanation:

`intsinx/(cos^2(x))dx`

using u-substitution:
let `u=cosx`. then `du=-sinxdx` or `-du=sinxdx`

substituting into the original integral:
`int-1/u^2du`

integrate that with power rule:

`1/u+C`

substitute `u=cosx`:

`1/cosx+C=secx+C`

Answer 2

`secx+C`

Explanation:

Based on the "inverse" nature of derivatives and antiderivatives (found through integrals), if we know that `d/dxsecx=secxtanx`, then `intsecxtanxdx=secx+C`.

Here, using `1/cosx=secx` and `tanx=sinx/cosx`, we see that

`intsinx/cos^2xdx=int1/cosx(sinx/cosx)dx=intsecxtanxdx=secx+C`