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$$\frac{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^2$$e^x$ - 2x$e^x$ + 2*$e^x$ + C = $e^x$($x^2$ - 2x + 2) + C.