SOLUTION: Give the integral of (2x)/(x^2+x-12) with respect to x.

Algebra ->  Trigonometry-basics -> SOLUTION: Give the integral of (2x)/(x^2+x-12) with respect to x.      Log On


   

Question 1182692: Give the integral of (2x)/(x^2+x-12) with respect to x.
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13209) About Me  (Show Source):
You can put this solution on YOUR website!


Use partial fraction decomposition.

2x%2F%28x%5E2%2B1-12%29+=+A%2F%28x%2B4%29%2BB%2F%28x-3%29

Multiply through by the least common denominator:

2x+=+A%28x-3%29%2BB%28x%2B4%29+=+%28A%2BB%29x%2B%284B-3A%29

Equate the coefficients of the linear and constant terms on the two sides of the equation to get two equation to solve for A and B.

A+B=2
4B-3A=0
3B+3A=6
7B=6
B=6/7
A=8/7

Rewrite the expression as the sum of two fractions with denominators that are linear polynomials:

2x%2F%28x%5E2%2Bx-12%29+=+%286%2F7%29%2F%28x-3%29%2B%288%2F7%29%2F%28x%2B4%29

Integrate using integral(1/(x-a))= log(x-a) plus a constant.

Answer by ikleyn(52879) About Me  (Show Source):
You can put this solution on YOUR website!
.

Many people,  when are asked about the integral of   1%2Fx,   mistakenly answer   ln(x) + Const.

This answer is  INCORRECT.

The  CORRECT  answer is   ln(|x|) + Const.



        And it works for  BOTH  positive and negative values of  x  (for all real values of  x  except of  0  (zero) ).



Correspondingly,   the integral of   1%2F%28x-a%29   is   ln(|x-a|) + Const1   (at x =/= a);

                            the integral of   1%2F%28x%2Bb%29   is   ln(x+b|) + Const2   (at x =/= -b).