linguistic:paraphrasing
LIATE rule

The LIATE rule is a rule of thumb for integration by parts. It involves choosing as u the function that comes first in the following list:

The function which is to be dv is whichever comes last in the list. The reason is that functions lower on the list generally have simpler antiderivatives than the functions above them. The rule is sometimes written as "DETAIL", where D stands for dv and the top of the list is the function chosen to be dv. An alternative to this rule is the ILATE rule, where inverse trigonometric functions come before logarithmic functions.

To demonstrate the LIATE rule, consider the integral

∫ x ⋅ cos (x) d x. {\displaystyle \int x\cdot \cos(x)\,dx.}

Following the LIATE rule, u = x, and dv = cos(x) dx, hence du = dx, and v = sin(x), which makes the integral become x ⋅ sin (x) − ∫ 1 sin (x) d x, {\displaystyle x\cdot \sin(x)-\int 1\sin(x)\,dx,} which equals x ⋅ sin (x) + cos (x) + C. {\displaystyle x\cdot \sin(x)+\cos(x)+C.}

In general, one tries to choose u and dv such that du is simpler than u and dv is easy to integrate. If instead cos(x) was chosen as u, and x dx as dv, we would have the integral

x 2 2 cos (x) + ∫ x 2 2 sin (x) d x, {\displaystyle {\frac {x^{2}}{2}}\cos(x)+\int {\frac {x^{2}}{2}}\sin(x)\,dx,}

which, after recursive application of the integration by parts formula, would clearly result in an infinite recursion and lead nowhere.

Although a useful rule of thumb, there are exceptions to the LIATE rule. A common alternative is to consider the rules in the "ILATE" order instead. Also, in some cases, polynomial terms need to be split in non-trivial ways. For example, to integrate

∫ x 3 e x 2 d x, {\displaystyle \int x^{3}e^{x^{2}}\,dx,}

one would set

u = x 2, d v = x ⋅ e x 2 d x, {\displaystyle u=x^{2},\quad dv=x\cdot e^{x^{2}}\,dx,}

so that

d u = 2 x d x, v = e x 2 2. {\displaystyle du=2x\,dx,\quad v={\frac {e^{x^{2}}}{2}}.}

Then

∫ x 3 e x 2 d x = ∫ (x 2) (x e x 2) d x = ∫ u d v = u v − ∫ v d u = x 2 e x 2 2 − ∫ x e x 2 d x. {\displaystyle \int x^{3}e^{x^{2}}\,dx=\int \left(x^{2}\right)\left(xe^{x^{2}}\right)\,dx=\int u\,dv=uv-\int v\,du={\frac {x^{2}e^{x^{2}}}{2}}-\int xe^{x^{2}}\,dx.}

Finally, this results in ∫ x 3 e x 2 d x = e x 2 (x 2 − 1) 2 + C. {\displaystyle \int x^{3}e^{x^{2}}\,dx={\frac {e^{x^{2}}\left(x^{2}-1\right)}{2}}+C.}

Integration by parts is often used as a tool to prove theorems in mathematical analysis.