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Problem 55 Hard Difficulty

Use a graph of $ f(x) = \frac{1}{(x^2 - 2x - 3)} $ to decide whether $ \displaystyle \int_0^2 f(x) dx $ is positive or negative. Use the graph to give a rough estimate of the value of the integral and then use partial fractions to find the exact value.


$$-\frac{1}{2} \ln 3 \approx-0.55$$


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Video Transcript

Let's use the graph of this function of to decide whether or not this integral from zero to two is positive. Negative. We'LL use the graft to give a rough estimate for the area or the integral. Then we'LL actually evaluating integral to find the exact So let's look at the graph of it. Here we have f graft and red, and let's focus on X values between zero and two. We see that the functions below that exacts is here. So in this case, then rule should be negative. Also, we can get a rough estimate to the area. So let's just approximated by taking this rectangle with side going from zero all the way down to negative point two five, and then the length will be from zero all the way to to. So in that case, we just have two times negative point two five. So that's just negative one half So we'll just say zero to FX. The X approximately We had negative point two five. So let's write that as a fourth times two, which was the link. This is negative one half, so that's that's a rough estimate. Now let's go ahead and actually evaluate this thing to find the exact value. That's where we'Ll need the methods of the section zero to. Now let's look at that denominator. We see that we can factor. This is X minus three X plus one. You should always do that. If you have a quadratic, always check whether or not that matters. Let's ignore the inner girl for one minute. Let's just look at the inside grant so that we can do our partial freshen the composition gin. So this is of the form a over X minus three. Be over X plus one. This is what the author would call Case won this thing to non repeatedly in your masters. Let's go and multiply both sides of this equation by the denominator on the left. Then we get one equals a X plus one plus be ex ministry, and we can rewrite this right hand side is by factoring out the X and then we're left over with a minus three B. So we see that there's no X on the left hand side, so we must have that a plus B equals zero. The constants are among the left is one that must mean that the constants are among the right as one. Let's go and solve this two by two system. If we subtract, we get four. B equals negative one. This means that bee is negative. One force No. So that is positive One fourth. Now we have our A and B Let's go ahead and plug these into the partial fraction to composition. And then we could integrate this from zero to and I'll go to the next page to start doing this. Yeah, the first inner girl zero two, Then a was one fourth X minus three. Then for the next term, we had negative wonder before for B, it's plus one D X. Now, if this minus three and plus one or bothering you, you can feel free here to do it yourself. In any case, we should end up with one of her floor. Then we have natural log X minus three for the first interval, zero to for the second and rule. We have a minus one over four and then natural log again. Now we just go ahead and simplify by playing in our end points. So for the first integral in parentheses here, if you plug into. Then you have negative one, but you take absolute value. You get positive one. And for the second term, you plugging zero. You get absolute value of negative three, which is three now for the second term, Plug in the two first and then plug in zero. Now does use the fact that natural log of one zero to cross out those terms. And then we can go ahead and combined these. We get negative two over four ln three, which is negative. One half Ellen three. That's the exact answer, and it's a negative number.

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