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integrate?

Posted: Sat Jun 30, 2012 6:19 am
by 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

Re: integrate?

Posted: Sat Jun 30, 2012 6:40 am
by 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.

Re: integrate?

Posted: Sat Jun 30, 2012 7:05 am
by 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?

Re: integrate?

Posted: Sat Jun 30, 2012 7:32 am
by vonLipwig
The derivative of sec(x) is sec(x)tan(x).

Re: integrate?

Posted: Sat Jun 30, 2012 7:56 am
by hadimotamedi
Thank you very much for your reply. I got the point clearly.

Re: integrate?

Posted: Sat Jun 30, 2012 8:25 am
by vonLipwig
No worries =)

Re: integrate?

Posted: Sun Jul 01, 2012 7:47 pm
by tcbh
So do you just have to memorize these trig integrals?

Re: integrate?

Posted: Sun Jul 01, 2012 8:38 pm
by 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.