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): A/(x-a)
• Repeated linear (x-a)^k: A1/(x-a) + A2/(x-a)^2 + ... + Ak/(x-a)^k
• Quadratic (x^2+px+q): (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.

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

Common JEE types:

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