Problem 47 Hard Difficulty

Make a substitution to express the integrand as a rational function and then evaluate the integral.

$ \displaystyle \int \frac{e^{2x}}{e^{2x} + 3e^x + 2}\ dx $

Answer

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Video Transcript

let's make a substitution to rewrite this fraction as a rational function. Let's take you to be you the X then do you is just either the ex d. X So we have. Do you equals you DX? Because either the X equals you and then saw for DX No. So when we've rewrite this integral, we can replace DX over here with do you over you. So let's just write that in now. So we don't forget that. And then here we have either the two X. But that's just use where So we have you squared in here. Then on the denominator, there's another East Square, then three, you plus two. No, Scott. And simplify this. We see that we could cancel one of the EU's. So we just have a one power up their leftover. So we just have you on top. Let's go and factor that denominator. That's you. Plus two and then you plus one for the quadratic money. Now, this is where we should do partial fraction to composition before we integrate. So let's go ahead and break you. You plus two, you plus one. This is what the author would call case one for partial fractions. It's when you have non repeated linear, indistinct polynomial is in the bottom. Now multiply both sides of this green equation by the denominator on the left, and you will get you a then you plus one just be you plus two and then go ahead and just simplify that right hand side. We can pull out of you and then we just have our constant term. A plus two b left over. So on the left hand side, the coefficient from the U. S. The one on the right. It's a plus B that gives us a plus. B equals one on the left, we see that the constant term zero there's no constant on the right. It's a plus two b, so that must be zero. And then we have a two by two system was going and subtract the green from the red. So when we do so, we negative B equals one or B equals negative one that's easier to work with and then saw for a and this equation up here two one minus B. So that's one minus negative one from here. And that equals two. So now we'LL just plug in our values for a nd into this partial fraction up here and we'LL integrate because we're still evaluating this red in anger over here. So started the next page. Since I'm running out of room, it's plugged in those values for Andy. They was, too. And then be was minus one. Let's just pull out that minus and then we have one over you plus one. Now we're ready to integrate. If this plus two and plus one are bothering you, feel free to use the use of pretties you can do, don't you? Plus two there. And for this one, you might help you to do w equals you plus one. In any case, to natural law, you plus two minus natural log absolute value, you plus one. And don't forget your constant sea of integration. And the last thing to do here is to use the definition of you to replace. You were eating eggs. So here have either the X plus two absolute value. Is love no longer needed because this is a positive number. Similarly, no absolute value over here is necessary and then plus see, and that's your final answer