# Integral

• Apr 12th 2011, 09:09 PM
joatmon
Integral
Evaluate:

$\displaystyle \displaystyle \int^{\frac{\pi}{2}}_0 [\frac{d}{dx} (\sin(\frac{x}{2}) \cos(\frac{x}{3})]dx$

Here is what I did:

$\displaystyle \displaystyle \int^{\frac{\pi}{2}}_0 \cos(\frac{x}{2}) \cos(\frac{x}{3})dx + \int^{\frac{\pi}{2}}_0 \sin(\frac{x}{2})(-\sin(\frac{x}{3}))dx$ (using the product rule to evaluate d/dx and separating the resultant integrals)

$\displaystyle \displaystyle \int^{\frac{\pi}{2}}_0 \cos(\frac{x}{2}) \cos(\frac{x}{3})dx - \int^{\frac{\pi}{2}}_0 \sin(\frac{x}{2})\sin(\frac{x}{3})dx$ (just rewriting to pull out the minus sign)

Assuming that this is right so far, I don't know how to evaluate these two integrals. I'm thinking that maybe I can use FTC to simplify this process somehow?

Thanks.
• Apr 12th 2011, 09:16 PM
pickslides
hmmmm,

consider $\displaystyle \displaystyle \frac{d}{dx}\left(\sin\left(\frac{x}{2}\right)\rig ht) =\frac{1}{2}\cos\left(\frac{x}{2}\right)$
• Apr 12th 2011, 09:45 PM
joatmon
You're right, but I'm still stuck. I just forgot to apply the chain rule on the derivatives, but all that does is adds a constant to each integral (I think):

$\displaystyle \displaystyle \int^{\frac{\pi}{2}}_0 \cos(\frac{x}{2}) \cos(\frac{x}{3})(\frac{1}{2})dx + \int^{\frac{\pi}{2}}_0 \sin(\frac{x}{2})(-\sin(\frac{x}{3}))(\frac{1}{3})dx$

$\displaystyle \displaystyle \int^{\frac{\pi}{2}}_0 \cos(\frac{x}{2}) \cos(\frac{x}{3})(\frac{1}{2})dx - \int^{\frac{\pi}{2}}_0 \sin(\frac{x}{2})\sin(\frac{x}{3})(\frac{1}{3})dx$

I don't see a U substitution here. Am I approaching this correctly?
• Apr 12th 2011, 09:51 PM
Prove It
Isn't the integral of the derivative of a function just that function itself?
• Apr 12th 2011, 10:06 PM
pickslides
Quote:

Originally Posted by joatmon

I don't see a U substitution here. Am I approaching this correctly?

You can continue this way, no u-sub needed, just use,

$\displaystyle \displaystyle \sin a \sin b = \frac{1}{2}(\cos (a+b)-\cos (a-b))$

$\displaystyle \displaystyle \cos a \cos b = \frac{1}{2}(\cos (a+b)+\cos (a-b))$

but it should match up with what Hayden has said in post #4
• Apr 12th 2011, 10:17 PM
joatmon
That's what I was wondering, but I am used to seeing it the other way around (with the derivative on the outside), so I didn't think that was legal. Thanks!
• Apr 12th 2011, 10:28 PM
joatmon
Sorry, but I am still confused. If the integral and the derivative "cancel each other out", then do I just evaluate the function at the upper limit like this:

$\displaystyle \displaystyle \sin(\frac{\pi}{4}) \cos(\frac{\pi}{6})$
• Apr 12th 2011, 10:35 PM
joatmon
I got it. That's the answer. To finish this off:

$\displaystyle \displaystyle \sin(\frac{\pi}{4}) \cos(\frac{\pi}{6})$

$\displaystyle \displaystyle \frac{\sqrt{2}}{2}\frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{4}$

Thanks, everyone.