When you use integration by parts, there are some common mistakes you should try to avoid:

Choosing $u$ and $dv$ the Wrong Way
Picking the right $u$ (the part you want to take the derivative of) and $dv$ (the part you want to integrate) is very important. A common mistake is choosing a $dv$ that is too complex. This can make the integral harder instead of easier. Usually, you want $u$ to be a polynomial (like $x^2$), a logarithmic function (like $\ln(x)$), or an inverse trigonometric function (like $\arctan(x)$). Choose $dv$ to be something simpler to work with.

Not Using the Integration Formula Correctly
The formula for integration by parts is: $\int u \, dv = uv - \int v \, du$
Make sure you use this formula the right way. If you forget a part or mix up the order, your answer might be wrong.

Forgetting About Constants
When you finish your integration, remember to add the constant of integration, $C$, to your answer. This small detail is important and forgetting it can cost you points on a test!

Not Looking for Recursion
Sometimes, after you apply integration by parts, you might end up with the same integral you started with. If this happens, you can solve the equation to isolate the integral. If you don’t notice this, it can make your work much harder.

Skipping the Check on Your Work
After you find the integral, it’s a good idea to check your work. You can do this by taking the derivative of your result to see if you get back to the original problem. This is an easy way to spot any mistakes in your calculations early.

By being careful about these common mistakes, you can make using integration by parts much easier and more successful!