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JH
Numerade Educator

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Problem 36 Medium Difficulty

Evaluate the integral.

$ \displaystyle \int \frac{1 + \sin x}{1 + \cos x}\ dx $

Answer

$-\cot x-\ln |\cot x+\csc x|+\csc x-\ln |\sin x|+C$

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Video Transcript

for this integral, luscious Tara. I re ready and then multiplying the numerator and denominator both. Bye, Let's do one minus co site. What So schools and simplify that. So multiply out numerator and the nominees are separately on the bottom one. Minus co sign squared Attack one minus cosa. Next, Plus I kn minus sign times Co sign running out of room here times Cosette Nix. And here we can rewrite. This is Science Square. That's one of your dragon identities. So it's good and split this into foreigner rules. What? So the first Herman over sine squared, then co sign science weird on the bottom for all these terms, then here we have sine X over science. Where do you see that? You could cancel one of those signs and then finally, our last term side Times co sign science Weird again. Cancel one of those signs here is Well, oh, so let's go to the next page and then simplify each of these fractions. That's the first thing, and we're off to re writing it. Second integral, you can go ahead and do a use of there if you want to, or you could just rewrite it as co tangent times because he can. Then the next one. This was the one over sign cause he can't minus. And then we have co sign over, Sign attention. And we know the answer to each of these these air just tricking the rules. So here, for the first one, we know it's negative. Attention X, then hear this week, plus course he can. And then over here we get a minus ln plus coach engine. And then over here we'LL have a minus natural log of sign. Let's add that constancy and and there's a final answer.