Evaluate the integrals in Exercises 9–28. It may be necessary ...

Question

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.

∫ [x / (x² + 4x + 3)] dx

Accepted Answer

Evaluate the indefinite integral to x + 1 divided by x2 + 3 x + 2DX. Now to solve this integral, we will make use of partial fraction decomposition. This will rewrite this function into a sum of simpler fractions. So to do that, we first need to factor our denominator. We have X2 + 3 X + 2. This factors to X + 1 multiplied by X + 2. We can then rewrite. A fraction We can write this then as a divided by x + 1 + B divided by x + 2. Now we need to find the coefficients of A and B. To do this, we can multiply both sides by our factor terms and simplify. We end up getting 2 x plus 1 equals a multiplied. By x + 2 + b multiplied by x + 1. Finally, we can expand by distributing A and B. We end up getting 2 x + 1 equals A X + 2A plus B X + B. We can then simplify this once more. By gathering our terms. We have X multiplied by a plus B. + 2 A + B Now, to solve this system, we will equate our coefficients. We noticed there's an X on the left and an X on the right. We'll use the coefficients here. We have 2 equals a plus b. And 1 equals 2 A plus B. Now we can simplify and solve. Let's use substitution by solving our first equation for b. We get B equals 2 minus A. We'll substitute this into our other equation. 1 equals 2a plus 2 minus a. This gives us a + 2 equals 1. Which gives us a equals -1. B then will be equals to 2 minus -1, which gives us 3. We now have our coefficients. We have -1 divided by X + 1 + 3 divided by x + 2. Now this is the integral we want to solve. Now, the integral. Of a function 1 divided by X plus A, it's just natural log absolute value X + A + C. So we can rewrite this. As negative natural log absolute value x + 1. Plus 3, natural log absolute value x + 2. Plus c Now, we can rewrite this once more. By putting our positive term first, we get 3 natural log absolute value x + 2 minus natural log X + 1 + C. This is the final answer for our integral. OK, I hope to help you solve it. Thank you for watching. Goodbye.

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.∫ [x / (x² + 4x + 3)] dx

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Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [x / (x² + 4x + 3)] dx