Partial Fractions

step 1:

integral (1 / (x²-1)) → a/(x+1) + b/(x-1)

integral (1 / (x-1)²) → a/(x-1) + b/(x-1)²

integral (1 / x²(x-1)) → a/(x-1) + bx+c/x²

step 2:

integral (1 / (x²-1)) → a/(x+1) + b/(x-1)

a(x-1)+b(x+1) / (x-1)(x+1)

1 = a(x-1)+b(x+1)

ex. integral(8/x(x+2)³)

a/x + b/(x+2) + c/(x+2)² + d/(x+2)³

(a(x+2)³+bx(x+2)²+cx(x+2)+dx) / x(x+2)³

step 3:

plug any numbers into x to solve for a and b

integral (a/(x+1) + b/(x-1))

note:

The integral at the start must have a degree in the numerator less than the denominator. If this isn’t the case, you must first divide the numerator by the denominator (ex. integral((x²-x+1) / (x²+x))

6x²+2 / x²+2x-3

6x²/x² = 6 (whole number)

6x²+0x+2 -6(x²+2x-3) = -12x+20 (remainder)

6 + ((-12x+20) / (x+3)(x-1))

yep

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