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Numerade Educator

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Problem 5 Medium Difficulty

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int_0^{\frac{\pi}{8}} \arctan 2x\ dx $

Answer

$$\frac{\pi}{8} \arctan \frac{\pi}{4}-\frac{1}{4} \ln \left(1+\frac{\pi^{2}}{16}\right)$$

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Video Transcript

Okay, This question wants us to evaluate in this integral using a table. So if we look at our table for something that looks like this closest thing we find is this formula right here that tells us what the anti derivative of inverse tangent of you d you is. And again, this could easily be derived using integration by parts. But since we have it in the table, we're just gonna take this form. So what we need to do is transformed are integral into this form. So to do that, we need to turn to X into you So we can do this, of course, using a use of So we just set you equal the two X and then to find our general factor, we just find that d u is to d x or 1/2 to you equals DX. And then we're almost done. We just got to transform our limits cause we have a definite integral. So you of pi over eight is two times pi over eight or pi over four. And then you of zero is two times zero, which is still zero. So now we can write our transformed into girl which is the integral from zero to pi over four of 1/2 tan inverse of you, do you? And of course, we can pull the 1/2 in front of the integral sign. So we get 1/2 times are anti derivative that we know from the table. So you tan inverse of you minus 1/2 times the natural log of one plus u squared. And we're evaluating this from zero sum pi over four. Okay, so now we can save ourselves some work here by noticing. But if we plug in our lower limit of zero, we get tan inverse of zero, which is zero minus 1/2 Ellen of one, which is also zero. So we just need to worry about the upper limit. So we get pi over four, divided by two times 10 in verse of pi over four minus 1/4 which I missed Here There's a 1/4 now Ellen off one plus pi over four. All squared, and we can drop the absolute value bars because we know this is bigger than one. So our final answer is high over eight times tan in verse of pi over four, minus 1/4. The natural log of one plus pi squared over 16 and there isn't anything we can do because remember Tan inverse of Pi over four. It's something we can't evaluate on Unit Circle while Tan of Pi over four is. So we have to leave it like that.