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JH
Numerade Educator

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Problem 35 Easy Difficulty

Evaluate the integral.

$ \displaystyle \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{x}{1 + \cos^2 x}\ dx $

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Video Transcript

So here let's define the int'l grant Bye after Lex. And let's note that f of negative X equals here we have negative X one plus co signs where negative X Now we could pull out this minus outside and then using the fact that co sign is even so here we were using the fact that Coulson is even we end up with negative FX. So this shows us that efforts are f is an odd function. So graphically, this means that the graph is is symmetric about the origin. So get this is not the graph of our function. But it does have this property that it symmetric about the origin. And then because if you look at her in points negative, Piper soon a pirate or two. So due to the cemetery around the origin, the a positive area that you get on one side cancels out with the negative area that's on the other. And this is coming from a theory, um, that we've seen earlier in the textbook that says that if you have a integral of this form, if if his are since we showed, f isn't our function, this in a girl has to be zero. And that's our final answer.