Partial Fractions Integral Involving the Natural Log Logarithm ln (Online calculus course test homework solutions)

Question: To 2 decimal places, what is the coefficient of the term involving ln(x-7) when we carry out the following integration for the following case:




Solution: There is no clear substitution for this problem and it definitely doesn't look like the derivative of a well known function. Since the denominator can be factored the method used will be partial fractions. Let's rewrite the integral:



We wish to rewrite this again so that it looks like a function we can integrate easily. So partial fractions allows us to split this into to fractions. So suppose the following equality holds. Then what would A and B have to be?




So we want our integral to look like the right side because that would be a simple logarithm once integrated. So let's solve for A and B by multiplying each side by (x-7)(x+2):




The tricky part about the final step is noticing that if the last equality is true, it MUST be the case that 16=(A+B), since they are the sole coefficients of x and 45=2A-7B since they are the constants. i.e.,



Now, by solving this system of equations we get that A=16-B which gives that 45=32B-7B=25B and that B=9/5. This means that A=16-9/5=71/5=14.2.

Thus, our integral becomes



Thus, the answer was merely finding the value A=14.2