In Exercises 9–16, express the integrand as a sum of partial f...

Question

In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ dx / (x² + 2x)

Accepted Answer

Welcome back everyone. Evaluate the integral dx divided by x2 + 3x. For this problem, let's consider the function that we're integrating, that's 1 divided by x2 + 3 x. What we're going to do is perform factorization, which gives us 1 divided by x x plus 3, and this allows us to use partial fraction decomposition. We're going to get a divided by the first factor, which is x, plus b divided by the second factor x + 3. And what we can do now is essentially find a common denominator. We are going to get a multiplied by x + 3 + b multiplied by x and the numerator divided by the common denominator x x plus 3. We have to recognize that our numerator is equal to 1, so we can now show that a multiplied by x + 3 plus b x is equal to 1, and we can expand this form to get a x plus 3 A plus b x equals 1. Additionally, we can perform factorization to show that A + B. Multiplied by x plus 3a is equal to 1, and this allows us to set a system of equations. First of all, we can show that a plus b must be equal to 0, and that's because we don't have any x terms on the right hand side, and 3a is going to be 1 because that's the free coefficient without x. From the 2nd equation we can show that a is equal to 1/3. And B is going to be A from the first equation. So B is equal to -1/3. Meaning we can now rewrite our integral as follows. We're going to have a divided by X, which is. 1/3 divided by x or basically 1/3 multiplied by 1 divided by x, right? That's exactly the same thing. And we're adding b divided by x plus 3. Specifically, we're subtracting. 1/3 multiplied by. 1 divided by x plus 3. And we're integrating with respect to x. We can now factor out 1/3 and integrate. 1 divided by x minus 1 divided by x plus 3. And these are elementary functions, right? We're going to get 1/3 in parenthes Ellen. The integral of 1 divided by x, the x is ln of the absolute value of x. And the integral of d x divided by x plus 3 is ln of the absolute value of x + 3. The same rule applies because we have a linear function in the denominator with a slope of 1, right? And now let's add the constant of integration c because that's an indefinite integral and simplify using the properties of logarithms which gives us 1/3 ln of the absolute value of x divided by. X + 3. Plus a constant of integration c. That's our final answer and thank you for watching.

In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.∫ dx / (x² + 2x)

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In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ dx / (x² + 2x)