When you start learning about integration in calculus, especially in Year 12 math, knowing the differences between substitution and integration by parts can help you solve problems better. Both methods have their own uses, but they work in different ways.

Substitution

Substitution is a technique that feels like working in reverse with the chain rule. You use it when one function is inside another, and you want to make the integral simpler. Here’s how it works:

1. Find a good substitution: Look for a part of the integral that you can replace with a single variable, like $u$.
2. Calculate $du$: Differentiate your substitution so you can express $dx$ in terms of $du$.
3. Rewrite the integral: Change your $x$ terms to $u$ terms to make the integral easier.
4. Integrate: Solve the integral using $u$.
5. Back substitute: Replace $u$ with its original form in terms of $x$ at the end.

This method is super useful when you see something that seems complicated but can be made simpler with a good substitution.

Integration by Parts

Integration by parts is handy when you're working with the product of two functions. It is based on a rule from differentiation and uses this formula:

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

Here’s a quick guide on how to do it:

1. Choose $u$ and $dv$: Pick parts of your integral wisely. Usually, let $u$ be a function that gets simpler when you differentiate it, and $dv$ is what's left over.
2. Differentiate $u$ and integrate $dv$: Find $du$ and $v$.
3. Use the formula: Put everything into the integration by parts formula.
4. Integrate again if needed: Sometimes you might need to integrate the new integral you get.

Key Differences

• Purpose: Use substitution for easier integrals when you can make a single change; use integration by parts for products of functions.
• Complexity: Substitution often makes the integral easier, while integration by parts can give you a new integral that might be harder but can still lead to the answer.
• Approach: Substitution is like simplifying things, while integration by parts needs a bit more thought in choosing $u$ and $dv$.

In the end, getting good at knowing when to use each method takes practice. Try working on problems using both methods, and soon you'll be able to pick the right one without thinking too hard!