Pattern 1: Finding f(x) from integral equation If integral(0 to x) f(t) dt = g(x), then f(x) = g'(x).

Pattern 2: Functional equations If integral(0 to x) f(t) dt = xf(x) + integral(x to 1) tf(t) dt: Differentiate: f(x) = f(x) + xf'(x) - xf(x). Simplify: 0 = xf'(x) - xf(x). So f'(x) = f(x), giving f(x) = $Ce^x$.

Pattern 3: Variable upper limit squared d/dx integral(0 to $x^2$) f(t) dt = f($x^2$) * 2x (chain rule on upper limit)

Pattern 4: Both limits variable d/dx integral(sin x to cos x) f(t) dt = f(cos x)(-sin x) - f(sin x)(cos x)