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

$ \displaystyle \int \frac{\sin x}{\cos^2 x - 3 \cos x}\ dx $

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

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 4

Integration of Rational Functions by Partial Fractions

Integration Techniques

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Let's make a substitution to express this. Given fraction as a rational function. Since we see that sign in the numerator, let's go ahead and choose you to be co sign then do you is negative. Sign XO Me, right, this's negative to you. Equal sign X d s. So then coming to our integral was pulled that minus sign from the EU. And then we have a deal. You up here that will give us the that negative. Do you gives us the sine x t x then in the bottom. This's just new squared minus to you by our definition of you. Now let's go ahead and we see a quadratic in the denominator. So before we ever do the partial fractions, we should always check that it's factored here we can pull out of you. And then there's leftover. You might history. Look over. Look now for a moment. Let's ignore the integral. Let's go to the side and write our friendship. Weaken right, this is a over you and then be over you minus three. This is what the author calls case. One distinct linear factors in the denominator, and they're not repeated. Let's multiply both sides of this equation by the denominator on the list when we do so, we have one on the left, a U minus three plus bu. Then let's pull out that you and there's our constant or minus three, eh? So on the left, we see that the constant term is one on the right, the constant her with no you that's minus three air. So we must have minus three equals one. And here on the right, the term in the constant in front of the U there next to the U. S. A. Plus B. But on the left is zero because there's no you. So we can go out and solve this two by two system. The first equation in red gives you a and then the second equation tells you that B is minus a. That's positive. Wonder. So we find our values for Andy was plug him into this partial fraction of composition appear and then will replace the fraction that we had with the partial fraction. And now we're ready to integrate. And don't forget that minus on the outside, I'm running out of room here, so let me go to the next stage negative. Then we have negative one third for a over you and then plus one third for B human history. Now, you may memorized how to do this interval the second one, even though there's a minus three there at that minus three is bothering you. Go ahead and do it. Use up on this case. Let me use the letter W and that should work you. So now watch out for the cancellation of the minuses here. So that's a plus one over three natural law give you and then that becomes a minus one over three natural log. And then this time you minus three plus c are constant of integration. The last step is just to replace you with co sign. That was the definition of you. Okay from the previous page. What? So play that in here and this equation becomes one over third Natural log. Absolute value co sign minus one over three natural log absolute value, then co sign X minus three, and then our constant of immigration. And that's your answer.

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