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JH
Numerade Educator

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Problem 58 Hard Difficulty

Evaluate the integral by completing the square and using Formula 6.

$ \displaystyle \int \frac{2x + 1}{4x^2 + 12x - 7}\ dx $

Answer

$\frac{1}{8} \ln |2 x-1|+\frac{3}{8} \ln |2 x+7|+C$

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Video Transcript

let's evaluate the general by completing the square and then using Formula Six. So looking at the denominator, it's good, in fact, around it for there from the first two terms, then leads the seven outside and then complete the square inside the parentheses. So then we'LL add in so three over to then we square that nine over four and then now we just by adding nine over four, we really added four times that over for which is nine and we make up for this by subtracting nine. Then we have four x plus three over to swear minus sixteen, which we correct. It is four square. So we've completed a square and let's go ahead and simplify this further. Let me go ahead and put the four inside the practices. And when we do that because of the square, it becomes the two. So we can do that. And then now we can rewrite this given problem two x plus one up, top the ex and then to have plus three squared minus four square. Well, we could try u substitution. Thank you to be too worthless. Torri, do you over too equals the X and this integral of top. We have two X plus one Well, such issue minus two. And we get that by subtracting both sides of this equation here by two we'LL get you minus two a few hours plus one and then we have to You don't forget the one half. Let's take that out on the bottom We see that we have use where and then minus four square. So let's go ahead and split this into two, two and a rules Phew! You slur minus four squared and then minus. Then we could cancel the twos. One over you squared minus four square. So we have two in a row is to evaluate and the one that I'LL circle run down red This is the one that will require formula Six. So we'LL get there in a few moments. Where is over here? The first Enter girl, we can do a u substitution. So let's go ahead and evaluate the cerebral separately. So here's the first integral and green for that integral here you should do what u substitution. Let's use a different letter w you swear? Minus four square or excuse me just in this case we'LL be back track here. I want you. W equals you square minus four square part of me, the whole denomination then dw to you, Teo So d w over too you Do you? Then? This integral is one half and then we have another one half from this integral DW over w So we have one over for Ellen. Absolute value. W and I am running out of room here. Let me go to the Paige. One over four, naturalized Absolute value w but using the definition of w that's just you squared minus four squared in absolute value. Let's not worry about the constant of integration because we still have another integral to compute. Well, I had the scene at the very end, using the definition of you from the original substitution National log Absolute value. Tuas plus three squared minus four square Absolute value. Now for the other liberals where one over you squared minus four square to you. Now let me write Formula Six. So for the six over here for Formula Six, this is from the textbook. Now, this is just one over to a natural log X minus a. It's plus a And then there will be our constancy. Yeah, and our problem we see is equal stood for So we have one over two times for natural log X minus four X plus four and then now, before we ad to see lashes and the answers to our previous and over which is up here. So let's go ahead and just write that final answer on a different color from the first Enter girl, we have one fourth natural log absolute value to Earth history Square minus four squared and then we have minus. This becomes a on the bottom So one over eight Natural law X minus four over X Plus four and I'LL Finally, let's go ahead and add that constant siham integration and that's your final answer.