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 x*$e^x$ dx: u = x (A), dv = $e^x$ dx (E) = x*$e^x$ - integral $e^x$ dx = x*$e^x$ - $e^x$ + C = $e^x$(x-1) + C

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

Case 3: Cyclic (Tabular) — returns to original integral $e^x$*sin(x) dx: Let I = integral. By parts twice: I = $e^x$*sin(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$\frac{1}{x}$dx = xln(x) - x + C Similarly: integral arctan(x) dx = xarctan(x) - $\frac{1}{2}$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.