Understanding how to use substitution and integration by parts can be tough for Grade 12 AP Calculus AB students. Each method has its challenges and works best in certain situations. Knowing when to use each method is important for doing well.

1. Substitution Method:

• The substitution method requires students to find a good replacement, often called $u$.
• Many students have a hard time figuring out what $u$ to pick that will make the math easier.
• For example, in the problem $\int x \cdot e^{x^2} \, dx$, choosing $u = x^2$ is very important, but beginners might miss this.
• If the chosen $u$ doesn’t help simplify things or if there are too many options, it can be frustrating.

2. Integration by Parts:

• This method uses the formula $\int u \, dv = uv - \int v \, du$.
• Students need to carefully choose $u$ and $dv$, or parts of their equation.
• If they make the wrong choices, it can lead to harder math problems or endless loops.
• For instance, in the integral $\int x \cdot \ln(x) \, dx$, if students don’t pick $u$ and $dv$ correctly, it makes solving much more complicated.

Even though these methods might seem overwhelming, practice helps a lot:

• Practice Problems: By working on more problems, students get better at knowing which method to use based on the type of function.
• Examples: Looking at examples that are already solved can help students figure out the right time to use substitution or integration by parts.

In the end, even though these techniques can be tricky, having a step-by-step approach and practicing regularly can really help students improve their skills in integration.