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

$ \displaystyle \int \frac{\sec^2 t}{\tan^2 t + 3 \tan t + 2}\ dt $

$$\ln |\tan t+1|-\ln |\tan t+2|+C$$

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 start by making the substitution here, let's take you to be tangent ot then do you? C can't square d t so this integral becomes D you over u squared three u plus two. Okay, and let's go ahead and factor mhm. And now we'll do partial fraction Shit. Let's go ahead and multiply both sides by this denominator here on the left and we get one equals a you plus one BU plus two and we can rewrite this. And by comparing coefficients, we see that a plus b has to be zero to be plus a has to be one. So we get a is minus one B as well. Let's go ahead and plug these in over here. And then we'll integrate this instead of integrating that. So plug again are constants. Yeah, And then there you can do a use sub here, for example, you can do and then here you can do you plus one, in any case, are integral becomes and that constant c and then finally recall that we defined you to be tangent of T. So we should go back here and plug this in. So this and there's our final answer

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