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Numerade Educator



Problem 62 Hard Difficulty

Evaluate the integral.

$ \displaystyle \int \frac{d \theta}{1 + \cos^2 \theta} $


$\frac{1}{\sqrt{2}} \tan ^{-1}\left(\frac{\tan \theta}{\sqrt{2}}\right)+C$


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Video Transcript

Let's start here by just multiplying topping denominator by C can't square. And then after multiplying, we just halve, seek and squared up top of the denominator. After multiplying we have she can't square data plus one now. It would be nice if we had a tangent in the denominator instead so that we can use the use up. But this is not equal to tangent. We have tents where they are, plus one equals sequence where data. However, we have a plus one here solicits ad once of both sides here. And after doing that, we just have seek and square data plus one. So we add one to the left side. So that's ten squared plus two. So this says we could come to our inner girl, keep the new brain residues and then rewrite that denominator as Tange is where plus two and then Now we're in shape to use the use of because if we let you be tangent when we do to you, that's where the sea can squared. Data is coming from. So this is our numerator and this animal can now be written as do you over Tan Square that's you square plus two, which, if you want you can also write. That is, do you over use were plus radical too square. So for this one you can use unless you memorize something in the table here, you could do it tricks up and you let you be square too Data. And after doing this use of so let's rewrite and a girl use were plus radical too square. So after integrating this, using the trips up that I just showed you, so are ten of you over Ruutu all over room too? Add that constant of integration in and then recall how we defined the original U substitution. I recall our original problem was stated in terms of data. So the last step here, it's just replace all those use when this case is just one you. That's a tangent, the numerator and then a radical two on the bottom, still dividing by this room, too at the constancy. And that's your final answer