Case 1: integral $sin^m$(x)*$cos^n$(x) dx

If m is odd: Save one sin(x), convert rest to cos(x) using $sin^2$ = 1-$cos^2$, substitute u = cos(x). Example: integral $sin^3$(x)$cos^2$(x) dx = integral (1-$cos^2$(x))$cos^2$(x)*sin(x) dx. Let u = cos(x).

If n is odd: Save one cos(x), convert rest to sin(x), substitute u = sin(x).

If both m and n are even: Use half-angle identities: $sin^2$(x) = (1-cos(2x))/2, $cos^2$(x) = (1+cos(2x))/2 Example: integral $sin^2$(x)$cos^2$(x) dx = integral $\frac{1}{4}$$sin^2$(2x) dx = $\frac{1}{8}$integral (1-cos(4x)) dx

*Case 2: integral sin(mx)cos(nx) dx (product-to-sum)

• sin(A)*cos(B) = $\frac{1}{2}$[sin(A+B) + sin(A-B)]
• cos(A)*cos(B) = $\frac{1}{2}$[cos(A-B) + cos(A+B)]
• sin(A)*sin(B) = $\frac{1}{2}$[cos(A-B) - cos(A+B)]

Case 3: integral $tan^n$(x) dx (reduction) integral $tan^n$(x) dx = integral tan^(n-2)(x)$tan^2$(x) dx = integral tan^(n-2)(x)($sec^2$(x)-1) dx = integral tan^(n-2)(x)*$sec^2$(x) dx - integral tan^(n-2)(x) dx = tan^(n-1)$\frac{x}{(n-1)}$ - integral tan^(n-2)(x) dx

Case 4: integral $sec^n$(x) dx For n even: save $sec^2$(x), convert rest using $sec^2$ = 1 + $tan^2$, substitute u = tan(x). For n odd: use by parts with u = sec^(n-2)(x), dv = $sec^2$(x)dx.

Case 5: Weierstrass Substitution (Universal) Let t = tan$\frac{x}{2}$. Then sin(x) = 2$\frac{t}{1+t^2}$, cos(x) = $\frac{1-t^2}{(1+t^2)}$, dx = 2$\frac{dt}{1+t^2}$. Converts any rational function of sin and cos to a rational function of t. Use as last resort — it often makes the algebra heavy.