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

1. ## 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

2. ## Re: Why are these two integrals not equal to each other?

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.

$\displaystyle \frac{\sin^2(ax)}{2a},~-\frac{\cos^2(ax)}{2a},~\frac{-\cos(2ax)}{4a}~$

3. ## 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?

4. ## Re: Why are these two integrals not equal to each other?

derp, I just realised.. Thanks for the help

5. ## 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.