Step 1: Is it a standard integral? Check the table of standard integrals first. Many JEE problems are direct applications.

Step 2: Is the derivative of the denominator (or part of it) in the numerator? If yes, the answer involves ln|denominator| or a power of the denominator.

Step 3: Is it a product of different function types? Apply integration by parts with LIATE rule. Check for $e^x$[f+f'] pattern.

Step 4: Is it a rational function? If deg(num) >= deg(den), do long division. Then apply partial fractions.

Step 5: Does it involve sqrt(quadratic)? Complete the square first, then use trig substitution or standard forms.

Step 6: Is it a trig integral? Identify the type (powers, products, rational in sin/cos) and apply the corresponding strategy.

Step 7: Can completing the square help? For any quadratic in the denominator or under a square root, complete the square to reach standard form.

Step 8: Last resort — Weierstrass substitution t = tan$\frac{x}{2}$ works for any rational function of sin and cos.

Verification: Always differentiate your answer to check. In JEE, if time permits, this step catches sign errors and missing factors.

Time Management: In JEE, integration problems average 3-4 minutes. If a method isn't working after 2 minutes, try a different approach. The $e^x$[f+f'] pattern and derivative-of-denominator checks take only seconds and solve ~30% of JEE integration problems.