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Numerade Educator

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Problem 14 Easy Difficulty

Evaluate the integral.

$ \displaystyle \int_0^1 \frac{dx}{(x^2 + 1)^2} $

Answer

$\frac{1}{4}+\frac{\pi}{8}$

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Video Transcript

So for this problem, we're gonna be using econometric identities to evaluate and and girls we know that what we're given is the integral from 0 to 1 of one over squared plus one spread the X. Okay, um and ultimately we can determine the final answer, but we want to do it in a more step by step process. So what we can dio is change X two k agents grade data plus one. Eso will change this whole thing to tangent squared data Awesome. And then this will change too. Seeking squared. OK, tha and then this can become data as a result. And this would change now, since we're moving in, terms of data is going to be a high over for then we can use a trigger metric identity. Um, because we know that this right here squared will give us seeking squared and then we square that it's going to become seek it to the fourth eso, Then you can cancel. So what we're left with is a one over seeking squared, and then, um, that's just going to obviously be a cosine squared. Then we can use another trig and metric identity, which is uh, that this is the same thing as because I kind of to theater plus one Andi that's founded by two. But the trick in Mexico, I didn't mean, um and that is the same thing as if we separate these fractions. It's just co sign of to fade over two plus one half D data. And that's the same thing as Thean a Gral of this, plus the integral of one half. Um, that's what that's ultimately going to give us is, um, something like this. We'll have one half sign of tooth Etem. You say that, then we'll have plus one half. It will be the liberal from zero to pi over four. Still Yeah, potato. I mean, you see that we keep getting the same an tries result, and this is us just simplifying things further each step. Now we can evaluate this. This will be the sign of tooth data, um, Times 1/4 and then evaluated at different values. Um, and what we ultimately get through the fundamental theme of calculus is that this is all the same as 1/4 plus high over eight. And as we see this is the same answer