When to use: Integrating P$\frac{x}{Q}$(x) where Q(x) factors into linear and quadratic terms.

Step 1: If deg(P) >= deg(Q), divide first to get polynomial + proper fraction.

Step 2: Factor Q(x) completely over the reals.

Step 3: Write the decomposition:

• Each factor (x-a) contributes $\frac{A}{x-a}$
• Each factor (x-a)^k contributes $\frac{A1}{x-a}$ + $\frac{A2}{x-a}$^2 + ... + $\frac{Ak}{x-a}$^k
• Each factor ($x^{2+px+q}$) contributes $\frac{Ax+B}{(x^2+px+q)}$
• Repeated quadratic factors: similar stacking

Step 4: Find coefficients by:

• Substituting roots of Q (cover-up method for distinct linear factors)
• Comparing coefficients of powers of x
• Using convenient values of x

Step 5: Integrate each term:

• $\frac{A}{x-a}$ => A*ln|x-a|
• $\frac{A}{x-a}$^n => A(x-a)^$\frac{1-n}{(1-n)}$
• $\frac{Ax+B}{(x^2+px+q)}$ => split into ln and arctan parts