Choosing the right functions for integration by parts can be a little challenging, but with some practice, you'll master it! This method is based on this formula:

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

Here are some simple tips to help you pick the right functions for $u$ and $dv$:

1. LIATE Rule: This is an easy way to remember what kinds of functions to choose. You should try to pick from these categories:

   • Logarithmic
   • Inverse trigonometric
   • Algebraic
   • Trigonometric
   • Exponential

   Always start from the top of this list. For example, if you have a logarithmic function, use that as $u$.

2. Simplification: When choosing $u$, make sure that $du$ (the derivative of $u$) is easier to work with than $u$ itself. This choice will make the new integral $\int v \, du$ simpler to solve.

3. Example to See It in Action: Let's look at the integral $\int x e^x \, dx$. You can set:

   • $u = x \quad \Rightarrow \quad du = dx$
   • $dv = e^x \, dx \quad \Rightarrow \quad v = e^x$

   Now, plug these into the integration by parts formula:

   $$\int x e^x \, dx = x e^x - \int e^x \, dx = x e^x - e^x + C$$

As you practice making these choices, integration by parts will become much easier!