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

1 Answer
Jun 27, 2018

int sec^2x(1+sinx)dx = tanx+secx+Csec2x(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 dxsec2x(1+sinx)dx=sec2xdx+sec2xsinxdx

The first integral can be solved directly:

int sec^2xd x = tanx +Csec2xdx=tanx+C

For the second integral note that:

int sec^2xsinx dx = int sinx/cos^2xdxsec2xsinxdx=sinxcos2xdx

int sec^2xsinx dx = int sinx/cosx1/cosxdxsec2xsinxdx=sinxcosx1cosxdx

int sec^2xsinx dx = int secx tanx dxsec2xsinxdx=secxtanxdx

This is also a well known integral:

int sec^2xsinx dx = secx+Csec2xsinxdx=secx+C

Putting it together:

int sec^2x(1+sinx)dx = tanx+secx+Csec2x(1+sinx)dx=tanx+secx+C