Step 1: Simplify the integrand. Use algebra (factoring, expanding, rationalizing), trigonometric identities (power-lowering, product-to-sum), or algebraic manipulation before choosing a method.

Step 2: Look for direct substitution. If a function and its derivative both appear, substitute. Common: integral f(g(x))*g'(x)dx => F(g(x)) + C.

Step 3: Check for special patterns.

• $e^x$[f+f']: answer is $e^x$*f + C
• Derivative of denominator in numerator: answer is ln|denominator| + C
• Exact derivative of some expression: direct antiderivative

Step 4: Classify the integrand.

• Rational P/Q: partial fractions
• Product of different types: integration by parts (LIATE rule)
• Powers of trig functions: reduction formulas or half-angle identities
• sqrt(quadratic): complete square + trig substitution
• Rational in sin/cos: check symmetry, then appropriate substitution

Step 5: If stuck, try:

• Weierstrass t = tan$\frac{x}{2}$ for trig integrals
• Reciprocal substitution x = 1/t for high-power denominators
• Euler substitution for sqrt(quadratic)
• Multiply by 1 in a clever form (conjugate, sec/sec, etc.)

Step 6: Verify by differentiation. Always check: d/dx[answer] should give the integrand.