How do you find ∫3x2−10x2−4x+4dx using partial fractions?

1 Answer
mason m
Feb 27, 2016

3x+12ln|x−2|−2x−2+C

Explanation:

First, since the degrees of the numerator and denominator are equal, use polynomial long division to rewrite the expression:

3x2−10x2−4x+4=3+12x−22x2−4x+4

Now, perform partial fraction decomposition on 12x−22x2−4x+4, recognizing that x2−4x+4=(x−2)2.

12x−22(x−2)2=Ax−2+B(x−2)2

Note that since the term is squared, it will be repeated.

Multiply both sides by (x−2)2 to see that

12x−22=A(x−2)+B

When we set x=2, we see that

12(2)−22=A(0)+B

2=B

Arbitrarily, set x=3 to solve for A, recalling that B=2:

12(3)−22=A(1)+2

A=12

Thus,

3x2−10x2−4x+4=3+12x−2+2(x−2)2

Now, we can integrate more simply:

∫3+12x−2+2(x−2)2dx

=3x+12ln|x−2|−2x−2+C

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