Raw Transcript
Hello Everyone, Tom from everystepcalculus.com and everystepphysics.com. Partial Fraction Decomposition. This time with three Partial Fractions.Let’s get started.Put index 8 in here with the open and closed parentheses, you get to my menu.You can scroll down to start up in the a section but you can scroll down alphabetical to Partial Fractions.Enter the integral by pressing Alpha first. Alpha, parentheses X squared plus 12 times X plus 12,close off the parentheses, divided by, open parentheses X cubed minus 4 times X. I always show you what you’ve entered. How many of you could factor this here? It’s pretty tough, I would think. Do it easy on my program. It comes up with 3 factors. So you write this in your paper. X squared plus 12x plus 12 is the numerator. Divided by the denominator times the denominator. And this is the actual denominator factored so we do the same thing here. And we’re going to use this to multiply times all of this to get rid of all the divisions. Common denominator is the actual words. So we added with x squared plus 12x plus 12 equals A and x times x minus 2 times x plus 2 Bx times x plus 2 Cx times x minus 2 And x equals 2. These quotation marks here is what the calculator does but you would put parentheses wherever you see these. parentheses 2 squared. You’re substituting 2 for every x in these functions. Use parentheses 2 here plus 12, etc. Do that throughout the whole system here. And that turns out to be B equals 5. At x equals zero. Use 0 for every x in all the computations here. And that turns out to be A equals minus 3 and x equals minus 2. Substitute for x in all the equation. C equals minus 1. Partial Fractions are minus 3 over x plus 5 over x minus two plus minus one over x plus two. And we integrate those. One thing about integrating a denominator with no, no exponent. You notice you can’t. This is an exponent of one, theoretically. But if you try to switch it to the numerator, five times X minus 2, it becomes a minus 1. Well in integration, you’re going to add one and then divide by the result. Well if you add one to a minus one, you get zero. In the numerator, anything to the zero is one. So that’s the reason you have to use logs. Whenever you see a no exponent to the denominator, it’s a log integration. So minus three logged minus the absolute X 5 log absolute x minus two minus one log x plus 2 plus C. Pretty neat, huh? everystepcalculus.com
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