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.

∫ (2x + 1) / (x² - 7x + 12) dx

Accepted Answer

Welcome back everyone. Evaluate the integral 3 x minus 2 divided by x2 minus 5 x + 6 dx. For this problem we're going to use partial fraction decomposition. Let's consider the function that we're integrating 3 x minus 2. Divided by x 2 minus 5 x plus 6, notice that we can factor out the denominator to get 3 x minus 2. Divided by X minus 2 multiplied by x minus 3. And now we're going to rewrite this expression as a divided by the first linear factor x minus 2 plus b divided by the second linear factor x minus 3. And now we're going to apply partial fractions. Let's find a common denominator, a multiplied by. X minus 3 + b multiplied by x minus 2 divided by X minus 2X minus 3. And now let's expand the numerator, which gives us A X minus 3A plus BX minus 2B. And this must be equal to the original numerator 3x minus 2. Now let's perform factorization, which gives us A + B multiplied by X minus 3A minus 2B. And this is equal to 3 x minus 2. And from here we can set a system of equations. A plus b must be equal to the coefficient in front of x, which is 3. And negative. 3 a minus 2b must be equal to -2, which is the free coefficient. Additionally, we can multiply both sides by -1 for the second equation to get positive signs, right? So 3a plus 2b equals 2. Now from the first equation we can show that b is equal to 3 minus a and we can substitute into the second equation which gives us 3 A plus 2 multiplied by 3 minus a equals 2. And therefore if we expand, we're going to get 3a plus 6 minus 2a equals 2. Combining the like terms, we get a plus 6 equals 2 and therefore we can subtract 6 from both sides to get a equals. -4. What is B? It's 3 minus A. So B equals 3 minus. -4, which is 7. Therefore we can rewrite our function as Considering the overall integral, we're going to add the integral sign, and then we're going to have a divided by x minus 2. So -4 divided by. X minus 2. Plus b divided by X minus 3, so plus 7 divided by x. -3 followed by dx. Now we can split our integral into two parts and factor out each constant, so that would be -4. Integral dx divided by x minus 2 + 7 integral dx divided by x minus 3. And those are elementary functions. Notice that in the denominator we have linear functions with a slope of one, so we will have. Natural logarithms as. The values of each integral for the first integral that would be -4 ln of the absolute value of the denominator x minus 2. And for the second integral that be plus 7 ln of the absolute value of the denominator x minus 3. Plus the constant of integration c because that's an indefinite integral. And that's our final answer for this problem. Thank you for watching.

In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.∫ (2x + 1) / (x² - 7x + 12) dx

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In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (2x + 1) / (x² - 7x + 12) dx