2–74. Integration techniques Use the methods introduced in Sec...

Question

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

12. ∫ (8x + 5)/(2x² + 3x + 1) dx

Accepted Answer

Welcome back, everyone. Determine the following integral integral of 5 X + 7DX divided by X2 + 4 X + 3. For this problem we're going to use partial fraction decomposition to integrate our function. Let's rewrite it. 5 x + 7 is divided by X2 + 4 x + 3. We can notice that our denominator can be factored. So we're going to get 5 X + 7 divided by. Rewriting our X2 + 4 X + 3 in its factored form, we get X + 1 multiplied by X + 3. And now using partial fraction decomposition, we can rewrite our expression as a divided by X + 1, our first linear factor, plus B divided by X + 3 our second linear factor. What we can do now is simply multiply A by x + 3. And we're going to add B multiplied by. X + 1 In other words, we're finding the common denominator, and we know that this sum must be equal to our previous numerator 5X + 7. Now expanding, we're going to get A X + 3 A plus B X plus B is equal to 5 X plus 7. And now combining our X terms, we get a plus B multiplied by X. Plus 3 A plus B, those are the three terms without X. We know that this is equal to 5x plus 7. And from here, we can set a system of equations. We can show that A plus B is equal to 5 because that's our coefficient adjacent to X. And then 3 A plus B must be equal to 7. That's our free coefficient. From the first equation we can show that B is equal to 5 minus A, and we can now substitute this expression into the second equation to get 3 A plus 5 minus a equals 7. Meaning 2 A equals 7 minus 5, which is 2. And therefore, A is equal to 1, while B is equal to 5 minus A. And 5 minus 1 is 4. So we can now show that we're integrating. A divided by X + 1 is 1 divided by X + 1. And B divided by X + 3 is 4 divided by X + 3. We're going to rewrite our integral as integral of 1 divided by X + 1. Plus 4 divided by X + 3D X. And now let's go ahead and integrate. Integrating 1 divided by X + 1, we get LN of the absolute value of X + 1. Plus, for the second part, we can factor out 4 and integrate the X divided by X + 3. The integral is going to be LN of the absolute value of X + 3. And we want to add the constant of integration C for an indefinite integral. That would be our final answer and thank you for watching.

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.12. ∫ (8x + 5)/(2x² + 3x + 1) dx

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2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
12. ∫ (8x + 5)/(2x² + 3x + 1) dx