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$$\text {Evaluate the following integrals.}$$$$\int_{0}^{5} \frac{2}{x^{2}-4 x-32} d x$$

Calculus 1 / AB

Calculus 2 / BC

Chapter 8

Integration Techniques

Section 5

Partial Fractions

Integrals

Integration

Applications of Integration

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Okay, We're going to start by decomposing our fraction here. So the X squared minus four x minus 32 does factor. Hopefully, you noticed that one of the factors is X minus eight, and the other factor is X plus four. So those are the denominators of our two fractions that are decomposition. So I'm going to say a over X minus eight plus b overexpose four will be the form. Then we multiply all that by our common denominator here to get rid of all the fractions. So the left hand side is just too. And we have each numerator times the opposite denominator because of the denominators reducing so eight times the X Plus four is a X plus for a B times the X minus eight B x minus eight b. And now we can equate our terms and the coefficients So there's no X. Well, that equals my hey plus B coefficients on the X and my constant of two legal my constant terms to equal for a minus eight b And then I just pick whatever method you want to use to solve that. And we end up with a equals positive 1/6 nb equals negative 1/6. So now we can rewrite this and I'm going to factor out my 16 because this would be 1/6 over the X minus eight and minus 1/6 over the X Plus four. So I'm just gonna factor out that common 1/6 and had 1/6 times the ending girl of one over X minus eight, minus one over X plus for DX. So now I can separate thes and integrate them individually. This is the one over you. Do you farm for an ankle formula? So this is really 1/6 helps. If I don't lay my arm on the such bad, this would be 1/6 times our integral on this is from 0 to 5. This is a definite integral. So we have the integral of one over X minus eight, which is the natural log of absolute value of X minus eight minus the natural log of the absolute value of expose four. And that's going to get evaluated from 05 so I can rewrite this as division just to make life a little simple. So 1/6 times in natural log of the absolute value of X minus eight over X plus four, and that's still being evaluated from 05 So now I plug in my five and my zero and subtract those so five minus eight over five plus four that's going to give me negative three overnight so that be the natural. Log off 1/3 when I take the absolute value minus the natural log of zero minus 8/0 plus four. Take the absolute value that is to. And so there we have our answer 16 times the natural world of 1/3 minus the natural log of to

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