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Evaluate the integrals.$$\int \frac{\sin \theta d \theta}{\cos ^{2} \theta+\cos \theta-2}$$

$\frac{1}{3} \ln \left|\frac{\cos \theta+2}{\cos \theta-1}\right|+C$

Calculus 2 / BC

Chapter 8

Techniques of Integration

Section 4

Integration of Rational Functions by Partial Fractions

Integration Techniques

Campbell University

Harvey Mudd College

Idaho State University

Lectures

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in this problem, we want to evaluate the integral of sine theta Genethia over coast Cynthia Coastline square Data plus Co sine theta minus you. All right, So we're gonna need this. Do a substitution. And the obvious choice seems to be you equal. Teoh co sign of data because of the top. We have a scientist, a dictator so on so that the U is equal to your negative sign of data data. So this becomes negative. Integral of do you Over you squared plus u minus you. Uh, now we have a fraction with upon a polynomial in the numerator and denominator so we can use partial fractions. Just put this up and make it easier to solve. So when we reverse boiler, you squared plus you mites to, uh, negative two is gonna be one times to the coefficient of you is positive one. So we want negative one rescues against you. Ah, positive one. Okay, so that's gonna be, um we wanna do partial fractions on U minus one times you plus to be some constant over a u minus one plus another constant over u plus two. Since both of these factors or linear, we can use the heavy side cover of method. Don't cover out u minus one. We'll have a is equal to 10 so few people is one. And you people think you so is gonna be one plus two, which was 1/3. As for B, we're covering up you plus two. So have won over negative two minus one, which is equal to negative 1/3 ready to rewrite the integral. So it's gonna be 1/3 time's one over U minus two. That's our A over U minus two minus 1/3 one over U plus three d You okay? So it wasn't like this Should be a two. Okay, All kinds around minus one plus two. All right, so now we know that's gonna be one through Ellen. Absolute value of U minus one minus 1/3 l An absolute value of you. Plus two plus a constant. We go ahead and combine these. So have 1/3 time's Ln absolute value of U minus one over absolute value of you. Plus you, plus a constant. So what was you You was equal to co sign of data. So this becomes 1/3 Ln absolute value of coastline, Athena, by this one over absolute value of co sign Saito. Plus you negative one is less or equal Teoh coastline. If data is less certain people, that one. So the numerator is gonna be, um, sending a positive or now yet numerator is gonna be a negative value. And the denominator is gonna be positive all the time so we can rewrite that. I wouldn't forget policy. So we can rewrite this as 1/3 Ellen of one minus co signing data over co sign of data plus two plus a constant and I'm blitz from.

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