The purpose of the cite is to encourage calculus students to go beyond the
book and develop good study habits. This semester, the blog will deal primarily with
the concepts of Calculus II.
Consider the problem of partial fractions. Assume that we have an integral involving a rational function which is quadratic in the denominator. We assume that the quadratic factors into distinct roots as shown.
It should be easy to verify that this formula works for all cases of distinct roots. Note that the divisor of the integral is just the distance between the two roots. The best technique is to write the roots in order with
"alpha" as the larger root. Then we know that the divisor should be positive.
It should be clear that the above formula takes care of all such cases. You should verify this by computing the derivative.
N.B. this technique is not intended
to replace methods where the student is required to show all their work rather it should allow the student to understand a simple method and to see the form of the solution for such a problem. The student should show whatever work their instructor requires for a complete solution.
Solve the following problems by observation using the formula technique.
ReplyDeleteanswers are
1) (1/3)LN(|x-5|/|x-2|) +c
2)(1/3)LN(|x-1|/|x+2|) +c
3)(1/2)LN(|x+3|/|x+5|)+c
4)(1/9)LN(|x-6|/|x+3|)+c