When you start learning about integrals, it can be confusing to know when to use substitution versus integration by parts. I've been there too, so let me help you understand when to use substitution.

1. Spotting Patterns

First, use substitution when you see something that looks like the "chain rule." This usually happens when one function is inside another one.

For example, if you have an integral like this:

$$\int f(g(x)) \cdot g'(x) \, dx,$$

substitution is a great choice! You can set $u = g(x)$, which makes the integral a lot simpler. If you see $g'(x)$, it’s a sign that substitution will make things easier.

2. Making the Integral Easier

Another reason to use substitution is if it helps simplify the integral itself. If you notice a polynomial or trigonometric function and replacing it will make the problem easier, go for it!

Take this example:

$$\int \sin(x) \cos^2(x) \, dx.$$

Here, if you change $u = \cos(x)$, the integral turns into a much simpler form:

$$\int \sin(x) \cdot u^2 \, dx,$$

which you can change to:

$$- \int u^2 \, du.$$

3. Trying Substitution First

It's often a smart idea to try substitution first. If you’re unsure which method to use, start with substitution. It's usually the easier choice and might be all you need. If it works, awesome! If not, then you can switch to integration by parts.

4. When to Use Integration by Parts

So when do you use integration by parts? You want to use this method when you have an integral that involves two different functions. It’s especially helpful if one function is easier to differentiate and the other is easier to integrate.

For example, with this integral:

$$\int x e^x \, dx,$$

this is the perfect time to use integration by parts. It’s all about picking the right method!

5. Practice Makes Perfect

In the end, using these techniques gets easier the more you practice. As you work on different types of integrals, you’ll get better at figuring out which method to use. It’s a bit of an art, but trust me, it will become second nature!

So remember, when you see a function that might need the chain rule, or when you can simplify the integral, think substitution. If you have a product of functions that fits the integration by parts method, go for that one. You’ve got this!