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JH
Numerade Educator

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Problem 80 Medium Difficulty

Evaluate the integral.

$ \displaystyle \int \frac{\sec x \cos 2x}{\sin x + \sec x}\ dx $

Answer

$\ln |\sin 2 x+2|+C$

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Video Transcript

So here let's just er off by multiplying topping denominator. Bye. Let's do it to co sign so many ways to do these types of inter girls. But sometimes multiplying top and bottom by the same sir can help simplify. And the reason I'm choosing co sign is to cancel out with the sea. Can't hear. So we recall that C can't is one of our coastline. So these terms here will cancel out. And so we see Jin and talked. We get too close I into x No t X still and then in the denominator we just have to sign eggs co signed X plus. And then when we multiply these two together, it's just too because of this fact Over here and just for convenience, let me rewrite this. It's just going to simplify things a little bit for us. Let me write. This is sign to X. I'm just using the double angle formula here and then let you be the denominator plus two. Then do you is using the chain rule. But you can see this to you is exactly the same bizarre numerator up here so we can rewrite this inaugural as one over you and then do you? That's just natural log of absolute value plus e. And we can write. This is natural Log signed two eggs plus two plus he and the reason absolute value is not needed here is because sinus between negative one we want so that what we add two side plus two is bigger than one. So in particular is a positive number, so we don't need absolute value here. And because of that, here's a final answer.