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

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Answer 1 · 2018-06-27T10:23:39

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

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#