Choosing u and dv: the ILATE rule

The problem of choice

The IPP works both ways: choosing u and dv poorly can complicate the integral rather than simplify it. An unfortunate choice can sometimes lead to a loop in the calculation or result in a more difficult integral than the original one.

The ILATE mnemonic rule

To prioritise the choice of u (the factor to be differentiated), we give preference to the following in this order: I – Inverse trigonometric functions (arcsin, arctan, etc.) L – logarithmic functions (ln x) A – algebraic functions (polynomials, x^n) T – trigonometric functions (sin x, cos x) E – exponential functions (e^x)

The highest factor in the list becomes u; the remainder (together with dx) becomes dv.

Example with ln(x)

Calculate J = the integral of ln(x) dx over [1, e]. There is only one visible factor: we set dv = dx (so v = x) and u = ln(x) (so du = dx/x), because ln x has no obvious antiderivative but a very simple derivative. J = [x ln x] from 1 to e – the integral from 1 to e of x * (1/x) dx = (e – 0) – the integral from 1 to e of 1 dx = e – (e – 1) = 1.

Example: a polynomial combined with a trigonometric function

For K = ∫ x² * cos(x) dx: A (algebraic x²) comes before T (trigonometric), so u = x² and dv = cos(x) dx.

A common pitfall

If we were to choose u = e^x and dv = x dx in I = ∫ x e^x dx, we would obtain v = x²/2, and the new integral ∫ (x²/2) e^x dx would be more complicated than the original: the degree of the polynomial increases rather than decreases. Always check that the new product obtained is simpler than the original one.