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# Prove the formula, where $m$ and $n$ are positive integers.$\displaystyle \int_{-\pi}^{\pi} \sin mx \cos nx dx = 0$

## If $m=n,$ then the first term in each set of brackets is zero.

#### Topics

Integration Techniques

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### Video Transcript

we'd like to prove that the inner world from negative pika pi of sign MX Times Co. Sign an X zero where Eman enter positive imagers. So to do this, let's take a look at them two grand. Let's call that F IX. So observe that half of negative X a sign and negative eggs co sign and negative X, or which weaken right is sign negative MX Coast Negative annex and then here for the sign we can pull out the minus sign. So this's sense sign is odd. So our equivalently this means a sign of negative T. It's negative, Sci fi and co sign is even so we can ignore the minus on the inside in just right it is co sign an ex. So these air properties of signing co signed that we know from Trig and we're left with up here We're left with negative f of X. So we conclude that eh facade So also we know that anytime you have ah odd function and you integrate it from negative eight a you'LL get a zero. So in this problem we haven't no girl from negative ada oven not function So the answer will always be zero and that proves the formula

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#### Topics

Integration Techniques

##### Heather Z.

Oregon State University

##### Kristen K.

University of Michigan - Ann Arbor

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