Hello Everyone, Tom from everystepcalculus.com and everystepphysics.com. Partial Fraction test problem. Index eight to get to my menu. Choose or school to Partial Fractions. I’m already there. To save time on the video. Press Alpha to enter any formulas into these entry lines on my programs. Alpha first. This one is alpha 5 divided by, alway parentheses. You can’t use too many as far as i’m concerned. Parentheses make everything clear. And certainly needed in division or fraction problems. Plus 3 times X minus 4. I always show you what you’ve entered. You can change it if you want. I say it’s okay.We factored the denominator, you noticed there’s two factors here. That means you’re going to be two partial fractions. So, you start by writing five which is the numerator divided by x squared plus 3x minus 4 times the denominator not factored yet. And then you to the partials. A divided by X minus one, which is the first factor and B divided by x plus 4, whic is a second factor. Always set that up. There is no x squares or anything so it’s very simple, here. Now when you multiply both sides by by the denominator you can eliminate a denominator in your left side. So you’re going to get just five and then course, when you do the multiplication in the denominator here,they switch around. A times x plus 4 plus b times x minus one. Now again, we try to we try to look where we can eliminate certain fractions here for factors. So you notice that minus 4 plus four zeros, it would limit this would it would make a minus five here. So anyways, at x equals minus 4. You get 5 is equal to, whenever you do these, you see quotation marks. I can’t put parentheses in the calculator. The way it works so. You’re gonna put parentheses here.Parentheses minus 4, parentheses plus 4. We are inserting this into the function so that you can do the math on it. Parentheses minus 4 parentheses minus 5. You can see B equals minus 5. So B equals minus 1. And same thing here at X equals one. This becomes zero B and this one becomes 5.So A equals one. Partial Fractions are one over x minus or plus -1 over X plus 4. And you do the integrals. You got 1 times log of x minus 1 absolute and minus 1 times log X plus 4 absolute plus C. Integration doesn’t become completed remember integration solves the original funtion, finds the area under the curve of the original function. But when there’s no limit or no range yet, then there’s always a plus C here.Plus constants. If we’ve had a range like from X equals 7 to X equals 10 or something, then we would come up with a number. And that number would be the area under the curve the original function. That’s all integration does, ever. Pretty neat, huh? everystepcalculus.com and everystepphysics.com
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