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JH
Numerade Educator

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Problem 59 Hard Difficulty

Evaluate the integral.

$ \displaystyle \int \frac{dx}{x^4 - 16} $

Answer

$$\frac{1}{32} \ln \left|\frac{x-2}{x+2}\right|-\frac{1}{16} \tan ^{-1}\left(\frac{x}{2}\right)+C$$

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Video Transcript

the first thing we should do here is such a cz factor that the not leader as much as we can. So here will get X squared minus four x squared plus four. And then we should check to see if he's fatter. So if we look at the first one here, this is just X plus two X minus two about the second one. Well, this one factor. So you look at the discriminatory here, B squared minus four a. C. It's a negative number here for this problem. For this x squared plus four. So that means it does not factor. So have to write it like that. And then we could go straight to the partial fraction so constant for the first two and then because we have irreducible contract IQ on the bottom and green, we have to put a linear up top. So we have CX plus de. So here, let me write this. We have to find a B C ity. Of course. So is And I got him on over thirty two. Be positive on over thirty two, and then we have that c zero and dia's minus one over eight, So I just pulled off the minus there. And then we have X square plus four. Now, the first two hundred girls, those air easier. And then we have one over thirty two l. A cops. Thirty two. They're not me. That's sloppy There. And then here we have X minus two. And then for this last inner girl here a little more difficult. In the first two, you could do a train from here, but sixty toothy data. And so when we integrate, this will have the one over eight with the minus from disturb right here and then after interbreeding. That's ten inverse of X over to and then divide by two again, all coming from the tricks up over here. Let me just go to the next page and write that out. So combining the log rhythms. So here, just combining log using the law of properties and then combining the the two that sixteen. And that's your final answer