How do you integrate sec(x)^2(1+sin(x))?

1 Answer
Jun 27, 2018

int sec^2x(1+sinx)dx = tanx+secx+C

Explanation:

Using the linearity of the integral:

int sec^2x(1+sinx)dx = int sec^2x dx + int sec^2xsinx dx

The first integral can be solved directly:

int sec^2xd x = tanx +C

For the second integral note that:

int sec^2xsinx dx = int sinx/cos^2xdx

int sec^2xsinx dx = int sinx/cosx1/cosxdx

int sec^2xsinx dx = int secx tanx dx

This is also a well known integral:

int sec^2xsinx dx = secx+C

Putting it together:

int sec^2x(1+sinx)dx = tanx+secx+C