*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 (1/4)*sin^2(2x) dx = (1/8)integral (1-cos(4x)) dx

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

• sin(A)*cos(B) = (1/2)[sin(A+B) + sin(A-B)]
• cos(A)*cos(B) = (1/2)[cos(A-B) + cos(A+B)]
• sin(A)*sin(B) = (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)(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(x/2). Then sin(x) = 2t/(1+t^2), cos(x) = (1-t^2)/(1+t^2), dx = 2dt/(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.