Partial fractions → Log Rule

A technique for integrating Rational functions.

▶ Jump back to previous note on Partial fractions.

▶Refer to Khan academy: Partial fraction expansion to evaluate integral

Example

This process is to break down the Rational Function to some simple fractions, which assume there are A & B leads to a system of equation:

• (A+B)·x + (B-A) = 1·x + (-4)

• So (A+B) = 1 and (B-A) = -4, which gets us A = 5/2 & B = -3/2

Strategy:

• Look at the Nominator & Dominator's degrees.

• If the dominator's degrees are higher or equal than the nominator, we do Long division of polynomial to downgrade it.

• Try to factorize the dominator if you can.

  • Assume two variables A & B

  • Apply the Partial Fraction Expansion technique.

• Apply the basic Log Rule to solve the parts.

Example

Solve:

Example

Solve: Refer to Symbolab.