Step 1: Ensure proper fraction. If deg(P) >= deg(Q), perform polynomial long division: P/Q = quotient + remainder/Q.

Step 2: Factor the denominator Q(x). Over the reals, Q factors into products of linear factors (x-a)^k and irreducible quadratic factors ($x^{2+px+q}$)^m where $p^{2-4q}$ < 0.

Step 3: Write the decomposition form.

• Linear (x-a): $\frac{A}{x-a}$
• Repeated linear (x-a)^k: $\frac{A1}{x-a}$ + $\frac{A2}{x-a}$^2 + ... + $\frac{Ak}{x-a}$^k
• Quadratic ($x^{2+px+q}$): $\frac{Ax+B}{(x^2+px+q)}$
• Repeated quadratic: stack with increasing powers in denominator

Step 4: Find coefficients.

• Cover-up: For distinct linear factor (x-a), set x=a after removing (x-a). Quick and efficient.
• Substitution: Plug in convenient x values (usually roots of Q).
• Coefficient comparison: Match coefficients of powers of x on both sides.

Step 5: Integrate each term.

• $\frac{A}{x-a}$ => A*ln|x-a|
• $\frac{A}{x-a}$^n (n>1) => A*(x-a)^$\frac{1-n}{(1-n)}$
• $\frac{Ax+B}{(x^2+px+q)}$: write Ax+B = $\frac{A}{2}$(2x+p) + (B-Ap/2). First part gives log, second gives arctan after completing the square.

Common JEE types:

• $\frac{1}{(x-a}$(x-b)): two distinct linear factors
• $\frac{1}{x(x^n+1}$): multiply by x^$\frac{n-1}{x}$^(n-1), then substitute t = $x^n$
• $\frac{px+q}{((x^2+a)}$($x^{2+b}$)): two quadratic factors