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 = $\frac{1-cos2x}{2}$, $cos^2$ x = $\frac{1+cos2x}{2}$

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 = $\frac{1}{2}$[sin(A+B) + sin(A-B)]
• sin A sin B = $\frac{1}{2}$[cos(A-B) - cos(A+B)]
• cos A cos B = $\frac{1}{2}$[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$\frac{x}{2}$cos$\frac{x}{2}$ or similar identities
• Last resort: Weierstrass substitution t = tan$\frac{x}{2}$

Strategy 5: Converting to tan$\frac{x}{2}$ Special useful case: $\frac{1}{a + b cos x}$ or $\frac{1}{a + b sin x}$ — use t = tan$\frac{x}{2}$ to get integral of rational function in t.