LIATE rule: Choose u (the function to differentiate) in this priority: Logarithmic > Inverse trig > Algebraic > Trigonometric > Exponential.

Tabular integration (repeated parts): For integral x^n * e^(ax) dx or x^n * sin(ax) dx, use a table of alternating derivatives of x^n and integrals of the other function, with alternating +/- signs.

Circular (self-referencing) parts: When parts leads back to the original integral. Example: I = integral e^x sin x dx. Parts twice gives I = e^x(sin x - cos x)/2 + C' - I. So 2I = e^x(sin x - cos x).

DI method (tabular): D (derivatives) | I (integrals) x^2 | e^x 2x | e^x 2 | e^x 0 | e^x

Result: x^2e^x - 2xe^x + 2*e^x + C = e^x(x^2 - 2x + 2) + C.