# Why are these two integrals not equal to each other?

• Mar 4th 2013, 09:58 AM
willsor
Why are these two integrals not equal to each other?
Hi

I have this integral:

integrate (sin(ax)cos(ax))

I used the identity that sinxcosx = 0.5sin(2x)

yet when I integrate (0.5sin(2ax)) it doesnt come out as the same answer as the original integral - at least not according to wolfram alpha's equality check-
Can anyone explain this?

Thanks

EDIT - a is just a constant
• Mar 4th 2013, 10:41 AM
Plato
Re: Why are these two integrals not equal to each other?
Quote:

Originally Posted by willsor
integrate (sin(ax)cos(ax))
I used the identity that sinxcosx = 0.5sin(2x)
yet when I integrate (0.5sin(2ax)) it doesnt come out as the same answer as the original integral - at least not according to wolfram alpha's equality check-

You may try any of these. They all work.

$\displaystyle \frac{\sin^2(ax)}{2a},~-\frac{\cos^2(ax)}{2a},~\frac{-\cos(2ax)}{4a}~$
• Mar 4th 2013, 10:46 AM
willsor
Re: Why are these two integrals not equal to each other?
Ok great - why can I not use this identity to make it easier in this case?
• Mar 4th 2013, 10:49 AM
willsor
Re: Why are these two integrals not equal to each other?
derp, I just realised.. Thanks for the help :)
• Mar 4th 2013, 05:22 PM
SworD
Re: Why are these two integrals not equal to each other?
When you integrate two equal expressions, you might get two different results, but they all have to vary by a constant. An example would be cos(x)^2 and -sin(x)^2. Don't forget to account for the constant of integration.