When we use the chain rule to take derivatives, there are some patterns that show up a lot. Some examples are
We can use these patterns to find derivatives.
The general strategy for integration by substitution has three steps:
- Change variables (substitute in u for some function of x).
- Apply an appropriate pattern to find the indefinite integral.
- Put the original variable back (substitute the function of x back in for u).
The trickiest part is usually figuring out which function we want to replace with u. Here are two guidelines that might help:
- u should be as complicated as possible, but still an "inside" function.
Practice:
Find the derivative of e4x. | |
Let u = 4x. Then we can rewrite: e4x = eu One of the patterns says (eu)' = eu ⋅ u'. Since u = 4x, we know u' = 4. We substitute these into the pattern and get (e4x)' = (eu)' = eu ⋅ u' = e4x ⋅ 4. We can also use these patterns to find antiderivatives. The equation (eu)' = eu ⋅ u' means the derivative of eu is eu ⋅ u'. Thinking backwards, this means
eu is an antiderivative of eu ⋅ u'.
We can write the family of all antiderivatives of eu ⋅ u' as eu + C. Using indefinite integral notation, 
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Find  . | |
The coefficient 4 is the derivative of the exponent 4x: 4e4x. Let u = 4x. Then u' = 4. We can rewrite the integral using u: 
Since multiplication is commutative, 
Applying the pattern, 
Finally, we put 4x back in for u to get e4x + C. We conclude that 
Rewriting the other patterns (assume a is a constant), we get 
We can use any of these patterns to find indefinite integrals. | |
Find  . | |
Let u = 5x2. Then u' = 10x. We'll be using the pattern 
We substitute u for 5x2, apply the pattern, then put 5x2 back in for u: 
In the example  we didn't choose u = x2. We chose u = 5x2 because it's more complicated, but still "inside" the cosine function: 
- u' (or some constant multiple of u') should already be a factor of the integrand.
In the example  we chose u = 5x2. Its derivative, 10x, was already a factor of the integrand: 
If the integral were 
instead (without the factor x), we couldn't choose u = 5x2 because its derivative u' = 10x isn't a factor of the integrand.
If the integral were 
instead (without the factor 10), we could choose u = 5x2 because the integrand includes the factor x, which is a constant multiple of u' = 10x. | |
For the integral, (a) identify u and u' and (b) integrate by substitution. 
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(a) We want u to be the most complicated thing that is still an "inside" function, so let u = 5x
u' = 5 (b) Let's go through the three steps. Change variables: 
Integrate using the appropriate pattern: 
and finally put the original 5x back in for u: -cos(u) + C = -cos(5x) + C. We have 
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For the integral, (a) identify u and u' and (b) integrate by substitution. 
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(a) The most complicated "inside" function is 3x + 2, so let u = 3x + 2. Then u' = 3. (b) To change variables, we break up the coefficient 15 into 5 ⋅ 3 so we can see where u' is: 
We undo the power rule to find the integral: 
and put 3x + 2 back in for u: (u)5 + C = (3x + 2)5 + C Our final answer is 
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For the integral, (a) identify u and u' and (b) integrate by substitution. 
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(a) The most complicated "inside" function is the exponent of e: 
We let u = 4x2 u' = 8x (b) We change variables, integrate using an appropriate pattern, and put the original variable back in: 
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Find  . | |
We have u = (x2 + 4) u' = 2x. The integrand does have the factor 2x hiding in it, as we can see if we break up the 16 into 8 ⋅ 2: 
From here we change variables, integrate, and put the original variable back in: 
Sometimes, however, we really do need to introduce a new factor to the integrand. To do this we use the trick of multiplying by a clever form of 1. | |
Find  . | |
We choose u = (x2 + 4). Then u' = 2x. We have the factor x in the integrand, but we're missing the 2. It's time for a trick. Multiply the integrand by 
Since  , we're not changing the value of the integrand. 
Now we do have 2x as a factor of the integrand, so we can change variables: 
Using one of the properties of integrals we can pull the constant  out of the integral. Then we can integrate and put the original variable back in: 
As our final answer we write 
You might reasonably wonder how the  turned into just + C. The expression 
accounts for the entire family of antiderivatives of (x2 + 4)7x. So does the expression 
Since both these expressions describe the same collection of functions, it doesn't matter which one we use. So we might as well use the simpler one and just write + C. | |
Find  . | |
Let u = 6x + 3. Then u' = 6. We don't have a factor of 6 in the integrand, but we can get one. First we multiply the integrand by  , which is a clever form of 1: 
We pull the factor  out in front of the integrand and leave the factor 6 inside: 
Now we can change variables and integrate: 
Similarly to the previous example, we write + C instead of  because either way we're describing the same family of functions. Here's a case where we can't use integration by substitution. | |
Find  . | |
The most reasonable thing to try is to let u = x2 + 4. Then u' = 2x. We have the factor 2 in the integrand. We might try to put a factor x in the integrand by multiplying by  : 
Now, however, we are stuck. We can't "pull out" the factor  to the outside of the integral. This doesn't work: PICTURE: put red slash, like no smoking sign, through the following equation: 
We can move constants back and forth across the integral sign, but not the variable of integration. When multiplying by a clever form of 1, we can take a slight shortcut. What we've been doing is multiplying the integrand by a clever form of 1 and then moving one factor outside the integral: 
Instead, we can immediately put the factor we want in the integrand inside the integral, and its reciprocal outside the integral: 
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For the integral, (a) identify u and u' and (b) integrate by substitution.
Answer
(a) The most complicated "inside" function is 6x + 4, so let
u = 6x + 4u' = 6
(b) First we change variables. We have to break up 30 into 5 ⋅ 6 so we can find u':
We use an appropriate pattern to integrate:
And finally put the original variable back in:
(u)5 + C = (6x + 4)5 + C
We conclude
For the integral, (a) identify u and u' and (b) integrate by substitution.
Answer
(a) The exponent of 4 is the most complicated inside function:
so we have
u = x2 + 4u' = 2x
(b) We change variables, integrate, and put the original variable back in:
For the integral, (a) identify u and u' and (b) integrate by substitution.
Answer
(a) The most complicated "inside" function is the ex inside the sin function:
The derivative of ex is ex, so
u = exu' = ex
(b) Change variables, integrate, and change variables back:
Integrate. The problem may or may not require substitution.
Answer
The most complicated function that's still inside something is the denominator of the fraction, which is "inside" the operation 
:
Let u = x + 1. Then u' = 1. We have
Integrate. The problem may or may not require substitution.
Answer
The most complicated inside function is the exponent of e:
Let u = sin x. Then u' = cos x.
Integrate. The problem may or may not require substitution.
Answer
This integral doesn't require substitution. It only requires thinking backwards. What has a derivative of -cos x? The answer is -sin x. So
Integrate. The problem may or may not require substitution.
Answer
We can rewrite this integral as
This makes it more obvious that we should choose u = sin x and u' = cos x.
Integrate. The problem may or may not require substitution.
Answer
The most complicated inside function is the exponent of 3, so let u = (2x2 + 4x) and then u' = (4x + x).
Integrate. The problem may or may not require substitution.
Answer
Let u = (2x + 3). Then u' should be 2. In order to see 2 as a factor of the integrand, we have to break up the coefficient 20:
Now we have to think backwards. What has a derivative of 10u4? Answer: 2u5. So
Integrate. The problem may or may not require substitution.
Hint
Simplify the integrand.
Answer
This doesn't require substitution. Simplify the integrand by raising 5 and x to the 4th power:
What has a derivative of 625x4? Answer:
So
Integrate. The problem may or may not require substitution.
Answer
If we only looked at the first guideline for how to choose u, we would take cos (ln x) as u since that's the most complicated thing that's still "inside" something else (it's the numerator of the fraction). However, the derivative of cos (ln x) is not a factor of the integrand, so that won't work. Let's try making u a little less complicated. Let
Then
Integrate. The problem may or may not require substitution.
Answer
Following the hint, rewrite the integral:
We have two choices. If we let u = sin x and u' = cos x, we get the integral
which isn't very helpful. On the other hand, if we let u = cos x, then u' = -sin x and we get an integral we know what to do with:
Integrate. The problem may or may not require substitution.
Answer
The most obvious choice of "inside" function is u = (3x2 + 4x + 7).
Then u' = (6x + 4).
Integrate.
Answer
Let
u = 3x + 4
u' = 3
We multiply by 3 inside the integral and 
without. Since 
we're not changing the value of the expression:
We integrate, writing C instead of 
, and put the original variable back in:
Integrate.
Answer
Let
In this case we need the factor 
inside the integral, and its reciprocal 4 outside the integral.
Integrate.
Answer
Let
u = e4x
u' = 4e4x
We need to introduce the factor 4 to the integrand, so we multiply the integrand by 4 and the outside of the integral by 
.
Those parentheses in the second-to-last step are important. If we had written
instead of
we would have gotten the wrong answer!
Integrate.
Answer
Let u = (3x2 + 4) and u' = 6x. Then we need to introduce 6 to the integrand and 
to the outside of the integral.
Integrate.
Hint
Factor 
.
Answer
Let u = (7x – 12) so u' = 7. We do have a factor of 7 in the integrand, which we can see if we factor 35.
Integrate.
Answer
Let 
. Then 
. We factor 
so we can see u'.
The other factor of 
isn't helpful, so let's move it outside of the integral:
The simplest antiderivative of u6⋅ u' is 
, so
Again we have 
and 
.
This time, we do want to move the factor 7 into the integrand since it's easy to find an antiderivative of 7u6u'.
Integrate.
Answer
Let 
. Since 
, we have
We rewrite the integrand:
This is almost what we want, but the sign is wrong. Since (-1)(-1) = 1, let's multiply the integrand by (-1)(-1). Then we can see u' in the integrand:
Integrate.
Answer
Let u = x2 + 4x and u' = 2x + 4. If we multiply the numerator of the integrand by 2, we get u'.
Integrate.
Hint
Answer
Following the hint, let's rewrite the integrand:
We choose u to be the denominator, because we know how to find the integral of 
but we don't know how to find the integral of 
. So let
u = sin(4x)
u' = 4cos(4x).
The integrand is missing a factor of 4, and we know how to fix that.
