Integration by Substitution A.K.A... The Reverse Chain Rule
Integration by substitution is just the reverse chain rule. If you learned your derivatives well, this technique of integration won't be a stretch for you.
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Let's say we want to find this integral:
For the sake of clarity, we´ll write it like this:
Now, what is the derivative of sinx? It is cosx. So, what we have inside the integral sign is a composite function and its derivative.
Here we'll use a very clever trick. We'll change variable. Let's invent a variable called u, that would be equal to sinx:
The derivative of u is:
If we substitute these two equations into the integral we get:
At this stage, you may not be very comfortable with differentials, but we can "cancel" them out:
So, now we have an integral we already know how to solve:
Finally, we just need to substitute u with sinx:
And that's all integration by substitution is about. Now, let's derive our answer to check it. This is something you can always do check your answers:
By the chain rule:
And this is the function we wanted to integrate!
What we did with that clever substitution was to use the chain rule in reverse.
We saw that the integral was probably a composite function. This is because, according to the chain rule, the derivative of a composite function is the product of the derivatives of the outer and inner functions.
And that's exactly what is inside our integral sign.
As a rule of thumb, whenever you see a function times its derivative, you may try to use integration by substitution.
Let's do some more examples so you get used to this technique.
Integration by Substitution Example 2
Let's find this integral:
Now, if you remember your derivatives, you know that the derivative of lnx is 1 over x.
To make clear why we need this fact, we'll write the integral like this:
So, again, what we have is a function and its derivative. How should we choose our u?
Remember that we will derive u. So, we should always choose u as the function whose derivative is inside the integral sign. That means:
So we have:
This is an integral we know:
Substitution back the u:
Integration by Substitution Example 3
Let's find the integral:
Will our u-substitution work here? Where is the derivative of the function there?
If we look closely, we'll note that we do have a composite function there.
We have the function ax. And then sin(ax), which is a composite function.
In our first example, we had a composite function and the derivative of the "inner" function. This integral is similar:
Can you see it? What is the derivative of x? It is one!
So, let's choose our u:
If we divide both sides of this equation by a, we get:
Now, we can substitute that 1 in our integral:
And we get:
This integral is easier to solve. What is the function whose derivative is sin(u)? It is -cos(u):
Now we'll do a tricky problem:
We can write this as:
We, again, have a composite function and its derivative. The harder part here is knowing how to choose our u:
So, we have:
If you look at your table of integrals, you'll find that:
This fact is easily proved using trigonometric substitution. So, finally we have:
- Integration by substitution can be considered the reverse chain rule.
- You'll need to know your derivatives well.
- Whenever you see a function times its derivative, you might try to use integration by substitution.
- With practice it'll become easy to know how to choose your u.
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