Lecture #7: Techniques of Antidifferentiation: Rational Functions & Partial Fractions

Techniques of Antidifferentiation

If \(p(x)\) and \(q(x)\) are polynomials (so that \(\frac{p(x)}{q(x)}\) is a rational function), it is always possible (in theory) to evaluate the antiderivative

\[ \int \frac{p(x)}{q(x)} \, dx. \]

However, doing so involves several algebra tricks, and potentially some substitution.

First Idea: Polynomial Division

Example: Evaluate \(\displaystyle\int \dfrac{x^3 + 1}{x^2 - 4} \, dx\).

Second Idea: Partial Fractions

Example: Evaluate \(\displaystyle\int \dfrac{4x + 1}{x^2 - 4} \, dx\).

Third Idea: Partial Fractions with Repeated Roots

Example: Evaluate \(\displaystyle\int \dfrac{4x + 1}{x^2 - 4x + 4} \, dx\).

Fourth Idea: Irreducible Factors

Example: Evaluate \(\displaystyle\int \dfrac{4x + 1}{x^2 + 4} \, dx\).

Fifth Idea: Several Irreducible Factors

Example: Evaluate \(\displaystyle\int \dfrac{4x + 1}{(x^2 + 4)(x^2 + 1)} \, dx\).

Sixth Idea: Irreducible Factors and Completing the Square

Example: Evaluate \(\displaystyle\int \dfrac{4x + 1}{x^2 + 2x + 4} \, dx\).

Seventh Idea: Repeated Irreducible Factors

Example: Evaluate \(\displaystyle\int \dfrac{(x + 1)(4x + 1)}{(x^2 + 4)^3} \, dx\).

Eighth Idea: Combining All of the Other Ideas

Example: Evaluate \(\displaystyle\int \dfrac{x^2}{x^3(x^2 + 1)} \, dx\).