This is one of the most valuable shortcuts that I use. It is advanced, but once you master this, integrals can become much, much faster. Nearly every integral that you come across after Calculus II requires U-Substitution, and that is why this trick is very useful.
Before we get to the good stuff, let’s do an example with regular U-Substitution.
Let’s try this again, but completely avoid U-Substitution. It’s a very similar concept, but it is much faster and will help you make less mistakes. With U-Substitution, you must remove yourself from the problem, go off to the side, figure out what u and du are, and then finally come back to the original problem. Working off to the side of a problem is where you can make a lot of simple mistakes, not to mention the mistakes that can be made while substituting back in. So how can we avoid this? At first, this may seem confusing or time consuming, but please trust me.
In U-Substitution, u is just a variable that stands for some function of x. u could be 2x, or it could be sin(7x +99). Why rename this function as u if we don’t have to? For just a second now, let’s think back to when you learned derivatives.
The derivative of 2x is equal to 2. This can also be written like this.
If you do some simple algebra, this can become
Another example of this:
Now , think about what this statement means. Any place you see 2·dx, you can replace it by d(2x), because they are equal.
Let’s take a look at the same example, and try to work through it with this new technique. At this point, you may not see where this is going, but you will soon.
Instead of using U-Substitution, we need to “build up” the differential (the dx). We need to change the differential to match the function we are trying to integrate.
What if we could make the integral look like this?
Wouldn’t this really be the same as below, but without the U’s?
The difference is that you do not need to take yourself away from the integral and deal with substitutions. You can work in the problem, not change variables, and avoid mistakes. Not only does this save time, but it also avoids a great deal of mistakes that you could very easily make. Ok, so let’s do the problem completely.
We want to change the differential to match the function you need to integrate. Ok, so if we want the differential to look like d(2x+5), then we need to know what d(2x+5) is equal to in terms of dx.
This part is just like U-Substitution. We have a dx in our original integral, but we do not have the 2. So what do you do? You create it! Just remember that if you put a 2 inside the integral, then you have to put a 1/2 on the outside to cancel it out.
Since we know that 2dx = d(2x+5),
Don’t actually write the very next step down, but please imagine that 2x+5 is like the variable ‘u’.
It is very important to see that this method is the exact same concept as U-Substitution. It works because of the same reasons. U-Substitution is great as an introduction because it is simpler to understand, but not worth it if you can understand this concept. I’ll do a few more examples without an explanation of the steps. Try it out for yourself. It works in any problem where you could use U-Substitution.
You don’t actually have to write the step saying what d(sinx) is equal to. That is just an extra visual to help you out. You can omit that step, and still be completely right. Just make sure you know what you’re doing when you do that, and not forget to account for the chain rule.
Another way to think of this concept is that you are moving things into the differential. Every time you move something, you must compensate for it so that the result remains equal to the original integral. For instance, if you need to put a 2 inside the differential, you must divide by 2 outside of it. If you have xdx, and you need d(x²), you can move the x into the differential. This works because d(x²) = 2x·dx. Just remember that in the process of moving the x into the differential, a 2 needed to go along with it. Therefore, you must put a 1/2 outside the integral t0 cancel out the 2. It is the same concept as U-Substitution, just a little more sophisticated.
I hope this trick helps you! Please post comments if anything is unclear. I can always edit the post to add more examples or explain something further. You can also send me a message and I will email you back with an answer to your specific problem. Thanks!