Methods from Rational functions

Given a proper rational function, each irreducible factor of the denominator contributes terms as follows.

• Linear factor $ax+b$: $\frac{A}{ax+b}$.
• Repeated linear factor $(ax+b)^k$: $\frac{A_1}{ax+b}+\frac{A_2}{(ax+b)^2}+\cdots+\frac{A_k}{(ax+b)^k}$.
• Irreducible quadratic factor $ax^2+bx+c$ with $a\ne0$: $\frac{Ax+B}{ax^2+bx+c}$.
• Repeated irreducible quadratic factor $(ax^2+bx+c)^k$: $\frac{A_1x+B_1}{ax^2+bx+c}+\frac{A_2x+B_2}{(ax^2+bx+c)^2}+\cdots+\frac{A_kx+B_k}{(ax^2+bx+c)^k}$.

To integrate a rational function on an interval containing no zero of its denominator:

1. If the function is improper, use polynomial division to write it as a polynomial plus a proper rational function and integrate the polynomial.
2. Decompose the proper part into partial fractions.
3. Integrate each linear-factor term using $u=ax+b$: its first power gives $\frac{A}{a}\ln|ax+b|$, while a power $k>1$ gives $\frac{A(ax+b)^{1-k}}{a(1-k)}$.
4. For $q(x)=ax^2+bx+c$ and a term $\int\frac{Bx+C}{q(x)^k}\,dx$, split the numerator into a multiple of $q'(x)=2ax+b$ and a constant remainder:

Then integrate the derivative term directly and complete the square in the constant remainder.