23-64. Integration Evaluate the following integrals.
35....

Question

23-64. Integration Evaluate the following integrals.

35. ∫ (x² + 12x - 4)/(x³ - 4x) dx

Accepted Answer

Hello there. Today we want to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Evaluate the interval, the interval of Y2 + 8 Y minus 15 divided by the power of 3 minus 9YDY. Awesome. So it appears for this particular problem we're ultimately trying to evaluate this particular integral. So with that in mind, our first step that we need to take in order to solve this particular prompt is we need to factor the denominator. So we need to take Y to the power of 3 minus 9Y and note that it is equal to Y multiplied by parentheses Y2 minus 9. Which is equal to Y multiplied by parentheses Y minus 3 multiplied by parentheses Y + 3, and that's when it's completely factored out. Fantastic. So once again, that's when we factor our denominator. Our second step now is we need to set up our partial fractions, so we need to take Y2 + 8 Y minus 15 divided by Y multiplied by parenthesis Y minus 3 multiplied by parenthesis Y + 3. Which will be equal to A divided by Y plus B divided by Y minus 3. Plus C divided by Y + 3. Fantastic. Our third step is we need to multiply both sides by Y multiplied by parenthesis of Y minus 3 multiplied by parentheses Y + 3 and equate our numerators. And when we do that, we will find that Y2 + 8 Y minus 15 is equal to a multiplied by parentheses Y minus 3 multiplied by parenthesis Y + 3. Fantastic. And then we need to add By multiplied by parentheses Y + 3 + CY multiplied by parentheses Y minus 3. Fantastic. Our fourth step is we need to expand the right side of our expression, and when we do, we will find that A will be multiplied by parentheses to the power of 2 minus 9 + B multiplied by parentheses 2 + 3Y plus C multiplied by parentheses Y2 minus 3 Y. Fantastic. Our 5th step is we now need to combine like terms, and when we do, we will get parenthesis A. Plus B + C multiplied by Y2 + 3 B minus 3C multiplied by Y minus 9A. Fantastic. Step 6, we need to match our coefficients. So when we match our coefficients, we will find that for Y2, the power 2, we will have A plus B plus C is equal to 1. And we will find that for Y, we will have 3B minus 3 C is equal to 8. Awesome. And for our constant, which I'll just simply denote as C, so for our constant C, we will have -9A, which is equal to -15. Fantastic. And for our seventh step, we need to solve for A, and when we solve for A, we will have -9A, which is equal to -15, which when we isolate, rather rearrange this expression to isolate sulfur A, you will find that A is simply equal to 5 divided by 3. Fantastic. Our 8th step, we need to substitute A equals 5 divided by 3 into our other equations. So when we do that, we will find that 5 divided by 3 plus B plus C is equal to 1, which we will find that B plus C is equal to 1 minus 5 divided by 3, which will be equal to -2 divided by 3. We'll also discover that 3B, so 3B minus 3C is equal to 8, which we will find that B minus C is equal to 8 divided by 3. Fantastic. So, moving right along here. Our ninth step is we need to solve the system. So we will find that B + C, so B + C is equal to -2 divided by 3. And B minus C is equal to 8 divided by 3. So with that in mind, we need to add the equations. So 2B is equal to 8 divided by, so 8 divided by 3 minus 2 divided by 3, which is equal to 6 divided by 3, which is equal to 2. So that will mean that B is equal to 1. Fantastic. And our step number 10 is we need to note that C will be equal to -2 divided by 3, minus 1, which will be equal to -5 divided by 3. Step 11. We need to write our integral as the integral of parentheses 5 divided by 3 Y plus 1 divided by Y minus 3, minus 5 divided by 3 multiplied by parentheses Y plus 3. Fantastic. Don't forget your parentheses DY. Fantastic. So now, Step 12 is we need to integrate each term, so we will take the interval of 5 divided by 3 Y D Y, which is equal to 5 divided by 3, multiplied by the natural log of the absolute value of Y plus C subscript 1, which I'll just call it C1 for short. Fantastic. Moving right along here. We now need to do our second integral, so the integral of 1 divided by Y minus 3 D Y, which will be equal to the natural log of the absolute value of Y minus 3 plus C2. And finally, we need to take the interval of -5 divided by 3 multiplied by parentheses Y + 3 D Y, which will be equal to -5 divided by 3, multiplied by the natural log of the absolute value of Y + 3 + C3. Fantastic. So, step 13 is we need to combine our results, so we need to take the integral of 5 divided by 3 Y. And let's to keep it more organized, let's say the interval of parentheses 5 divided by 3 Y plus 1 divided by Y minus 3, minus 5 divided by 3 multiplied by parenthesis Y plus 3. And don't forget your parenthesis and D Y. Fantastic, which this will be all equal to 5 divided by 3, multiplied by the natural log of the absolute value of Y plus the natural log of the absolute value of Y minus 3, minus 5 divided by 3, multiplied by the natural log of the absolute value of Y + 3 + C. Fantastic. So we could take it a step further and we could write that the integral of parentheses 5 divided by 3 Y plus 1 divided by Y minus 3 minus 5 divided by 3 multiplied by the parenthesis of Y plus 3. Parenthesis D Y will be simply equal to 5 divided by 3 multiplied by the natural log of the absolute value of Y divided by Y + 3. Plus the natural log of the absolute value of Y minus 3. Plus C, and that will be our final answer. Hooray, we did it. So that means going back to look at the problem itself to recap what we've learned, that will mean that once again, our final answer, without a doubt, will have to be the following. So, once again, our final answer is 5 divided by 3, multiplied by the natural log of the absolute value of Y divided by Y + 3 plus the natural log of the. The value of Y minus 3 + C, noting once again that C represents some constant value. And that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

23-64. Integration Evaluate the following integrals.35. ∫ (x² + 12x - 4)/(x³ - 4x) dx

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23-64. Integration Evaluate the following integrals.
35. ∫ (x² + 12x - 4)/(x³ - 4x) dx