Strategy 1: Powers of sin and cos (sin^m x cos^n x)

• One odd power: save one factor, convert rest via sin^2+cos^2=1, substitute
• Both even: use half-angle formulas sin^2 x = (1-cos2x)/2, cos^2 x = (1+cos2x)/2

Strategy 2: Powers of tan and sec

• integral tan^n x dx: use tan^2 x = sec^2 x - 1 to reduce
• integral sec^n x dx: for odd n, use by parts; for even n, save sec^2 x for du

Strategy 3: Products sin(mx)cos(nx)

• sin A cos B = (1/2)[sin(A+B) + sin(A-B)]
• sin A sin B = (1/2)[cos(A-B) - cos(A+B)]
• cos A cos B = (1/2)[cos(A-B) + cos(A+B)]

Strategy 4: Rational in sin x, cos x

• Numerator is derivative of denominator: gives ln|denominator|
• Use sin x = 2sin(x/2)cos(x/2) or similar identities
• Last resort: Weierstrass substitution t = tan(x/2)

Strategy 5: Converting to tan(x/2) Special useful case: 1/(a + b cos x) or 1/(a + b sin x) — use t = tan(x/2) to get integral of rational function in t.