Question

X^3/(x-1)(x-2) by partial fraction

Answer 1

Answer:

=−16ln|x−1|+115ln|x+2|+110ln|x−3|+C

Explanation:

Using partial fraction decomposition , break up the fraction into three different fractions added together:

∫1(x−1)(x+2)(x−3)dx

=∫(Ax−1+Bx+2+Cx−3)dx

−−−
In order to find the values of A,B,andC, use common denominators, and set it equal to the original fraction.

A(x+2)(x−3)+B(x−1)(x−3)+C(x−1)(x+2)=1

Plug in any value for x. The easiest numbers to plug in are those which make one of the factors zero, because many terms cancel:

If x=1: A(1+2)(1−3)=1
A=−16

If x=-2: B(−2−1)(−2−3)=1
B=115

If x=3: C(3−1)(3+2)=1
C=110
−−−
The integral after plugging in values of A,B,andC becomes:
=∫(−16x−1+115x+2+110x−3)dx

=−16ln|x−1|+115ln|x+2|+110ln|x−3|+C