integrate?

[hadimotamedi]

Dear All
I need to compute the following integral :
f(x)=(1+sin(x))/((cos(x))^2)
I=integral(f(x)dx)
Can you please propose a simple change of variable to simplify its calculation?
Thank you

[vonLipwig]

You don't need to change variables, just break it up into two fractions (one for each term in the numerator).

Then it comes down to integrating 1/(cos^2) and sin/(cos^2), or, if you prefer, sec^2 and sec*tan. Those are integrals you're probably familiar with.

[hadimotamedi]

Thank you very much for your help. So the integral breaks into (sec(x))^2 + sec(x).tan(x)
For the fist term, it is straightforward but what about the second term?

[vonLipwig]

The derivative of sec(x) is sec(x)tan(x).

[hadimotamedi]

Thank you very much for your reply. I got the point clearly.

[vonLipwig]

No worries =)

[tcbh]

So do you just have to memorize these trig integrals?

[vonLipwig]

The ones we just used are pretty basic, so yes, I'd remember them, as well as the fact that those for cosec and cot look similar but with some minus signs. Anything more complicated, I'd substitute for sin, cos or tan and try and work it out that way, rather than trying to remember.