When students learn about integration by parts, they often run into some common mistakes that can slow them down.

First, many struggle with choosing $u$ and $dv$ the right way. The choice should follow something called the LIATE rule. LIATE stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential. Picking the wrong option can make the problem harder instead of easier.

Next, students sometimes make mistakes when calculating $du$ and $v$. When you differentiate and integrate, even a small error can lead to the wrong answer. So, it’s really important to pay attention to numbers and signs during these steps.

Another common mistake is forgetting to add integration constants when working with indefinite integrals. If you skip this, you might end up with the wrong conclusion about the integral’s value.

Additionally, students often misuse the formula itself. The correct integration by parts formula looks like this:

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

If you leave out any part of this equation or use it incorrectly, it can lead to confusion or wrong answers.

Finally, students sometimes forget to check their answers by differentiating their results. This step helps to make sure everything matches up with the original problem.

By being aware of these common mistakes, students can get better at using integration by parts.