Integration by Parts

Integration by Parts is a method of integration used for integrands which consist of a product of two or more functions.

Formula

$$\int u dv = uv - \int v du$$

Choosing $u$ and $dv$

The choice of which functions to assign to the $u$ and $dv$ variables is the most critical decision you'll make while using the Integration by Parts method.

In general:

Choose $dv$ to be something that's easy to integrate.
Choose $u$ to be something that becomes more simple, or "smaller," when you take the derivative.

Professor Teague's Shortcut for Choosing $u$

Professor Teague taught us an acronym which is extraordinarily useful in helping you to easily determine which function should be $u$. Unless you're integrating a product of 3 or more functions, then once you've chosen the $u$ function, all you're left with is $dv$.

The shortcut is based on an acronym:

LIATE

L = Natural Logarithm
I = Inverse Trigonometric Functions
A = Algebraic Functions
T = Trigonometric Functions
E = Exponential Functions

To use this method of choosing $u$ and $dv$, you examine the integrand, while reading the acronym from left to right. In that order, choose the $u$ function.

For instance, if you were asked to integrate: $$\int x^2e^x dx$$

$x^2$ is an algebraic function, while $e^x$ is an exponential function. Because algebraic functions come before exponential functions in the acronym—reading it left-to-right—the choice of function for $u$ is obvious: it must be $x^2$.

Similarly, in the following integrand, $$\int x\arctan(x) dx$$

you can see that $\arctan(x)$ (or $\tan^{-1}(x)$—the alternate notation) is an inverse trigonometric function, which appears before the algebraic functions in the LIATE, you should assign the function $x$ to $u$.