Formula: integral u dv = uv - integral v du

LIATE Rule for choosing u: L — Logarithmic (ln(x), log(x)) I — Inverse trigonometric (arcsin, arctan, etc.) A — Algebraic (x, x^2, polynomials) T — Trigonometric (sin, cos, tan, etc.) E — Exponential (e^x, 2^x, etc.)

Choose u as the function that appears earliest in LIATE.

Case 1: Single Application integral xe^x dx: u = x (A), dv = e^x dx (E) = xe^x - integral e^x dx = x*e^x - e^x + C = e^x(x-1) + C

Case 2: Repeated Application integral x^2sin(x) dx: Apply by parts twice. First: u = x^2, dv = sin(x)dx => x^2(-cos(x)) - integral (-cos(x))2x dx = -x^2cos(x) + 2integral xcos(x)dx Second: u = x, dv = cos(x)dx => xsin(x) - integral sin(x)dx = xsin(x) + cos(x) Final: -x^2cos(x) + 2xsin(x) + 2cos(x) + C

Case 3: Cyclic (Tabular) — returns to original integral e^xsin(x) dx: Let I = integral. By parts twice: I = e^xsin(x) - e^x*cos(x) - I So 2I = e^x(sin(x) - cos(x)), giving I = e^x(sin(x) - cos(x))/2 + C

Case 4: Single Function by Parts integral ln(x) dx: u = ln(x), dv = dx = xln(x) - integral x(1/x)dx = xln(x) - x + C Similarly: integral arctan(x) dx = xarctan(x) - (1/2)ln(1+x^2) + C

Case 5: Tabular Method (Repeated by Parts) For integral x^n * e^(ax) or integral x^n * sin(ax): Create a table alternating derivatives of u and integrals of dv with alternating signs (+, -, +, -, ...). Multiply diagonally and sum.