Editing Integration by parts (section)

====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:<ref>{{Cite journal |jstor=2975556 |first=Herbert E. |last=Kasube |title=A Technique for Integration by Parts |journal=[[The American Mathematical Monthly]] |volume=90 |issue=3 |year=1983 |pages=210–211 |doi=10.2307/2975556}}</ref>
* '''L''' – [[logarithmic function]]s: <math>\ln(x),\ \log_b(x),</math> etc.
* '''I''' – [[inverse trigonometric function]]s (including [[Inverse hyperbolic functions|hyperbolic analogues]]):  <math>\arctan(x),\ \arcsec(x),\ \operatorname{arsinh}(x),</math> etc.
* '''A''' – [[algebraic function]]s (such as [[polynomials]]): <math>x^2,\ 3x^{50},</math> etc.
* '''T''' – [[trigonometric functions]] (including [[Hyperbolic functions|hyperbolic analogues]]): <math>\sin(x),\ \tan(x),\ \operatorname{sech}(x),</math> etc.
* '''E''' – [[exponential function]]s: <math>e^x,\ 19^x,</math> etc.

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 [[antiderivative]]s 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

<math display="block">\int x \cdot \cos(x) \,dx.</math>

Following the LIATE rule, ''u'' = ''x'', and ''dv'' = cos(''x'') ''dx'', hence ''du'' = ''dx'', and ''v'' = sin(''x''), which makes the integral become
<math display="block">x \cdot \sin(x) - \int 1 \sin(x) \,dx,</math>
which equals
<math display="block">x \cdot \sin(x) + \cos(x) + C.</math>

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

<math display="block">\frac{x^2}{2} \cos(x) + \int \frac{x^2}{2} \sin(x) \,dx,</math>

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

<math display="block">\int x^3 e^{x^2} \,dx,</math>

one would set

<math display="block">u = x^2, \quad dv = x \cdot e^{x^2} \,dx,</math>

so that

<math display="block">du = 2x \,dx, \quad v = \frac{e^{x^2}}{2}.</math>

Then

<math display="block">\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 x e^{x^2} \,dx.</math>

Finally, this results in
<math display="block">\int x^3 e^{x^2} \,dx = \frac{e^{x^2}\left(x^2 - 1\right)}{2} + C.</math>

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