Partial Fractions Calculus II

Question

Please help me understand how to find the integral of (1)/(x^3+2x^2+3x). Test is tomorrow pls help.

Answer 1

A rational fraction may be written as a sum of partial fraction as follow:

1/[x(x-1)(x+3)] ≡ (A/x) + [B/(x-1)] + [C/(x+3)]

This equation must be IDENTICALLY true which means that the coefficients of each power of x must be equal in order that the equation be true for all values of x.

So now clear fractions (and ignore the denominator) so that you get

Ax² + 2Ax + 3A +

Bx² - Bx +

Cx² + 3Cx

which gives you

A+B+C=0

2A-B+3C=0

3A=1

then A= 1/3

Substitute to get 2 equations in the unknowns B and C...and solve

Then each one of the fractions in the second line above will integrate as a ln which you can then collect.