complex trig/integration: int (sinx + sinx tan^2x)/(sec^x)dx

[crzymath]

definite integral
integral [(sin x + sin x tan[sup:2hagkdeu]2[/sup:2hagkdeu]x)/(sec[sup:2hagkdeu]2[/sup:2hagkdeu]x)] dx

here is what i did:
integral [(cos x + sin x tan[sup:2hagkdeu]2[/sup:2hagkdeu]x)/(sec[sup:2hagkdeu]2[/sup:2hagkdeu]x)] <--- i didnt know how to break down the other trig identities.

 

[mmm4444bot]

sin(x) = cos(x) is NOT an identity.

You may not replace sin(x) with cos(x)!

 

[crzymath]

Re: complex trig/integration

doesnt tan[sup:289j7kod]2[/sup:289j7kod]x = sec[sup:289j7kod]2[/sup:289j7kod]x-1 <--- can i use that somewhere?

 

[mmm4444bot]

Yes, and yes.

Factor out sin(x) from the numerator. Use the identity to cancel the remaining factor with the denominator.

 

[crzymath]

Re: complex trig/integration

ok here is what i did:

problem:
integral (sin x + sin x tan[sup:1fxk5tlh]2[/sup:1fxk5tlh]x)/(sec[sup:1fxk5tlh]2[/sup:1fxk5tlh]x) dx

=integral sin x (1 + tan[sup:1fxk5tlh]2[/sup:1fxk5tlh]x)/(tan[sup:1fxk5tlh]2[/sup:1fxk5tlh]x + 1)
=integral sin x <--- the tan[sup:1fxk5tlh]2[/sup:1fxk5tlh]x cancels out
=-cos x + C