For the integral, (a) identify u and du and (b) integrate by substitution.
Answer
(a) The biggest thing that's inside something else is 3x, so let u = 3x. Then
du = 3dx
(b)
Example 2
For the integral, (a) identify u and du and (b) integrate by substitution.
Answer
(a) We have
u = 6x
du = 6dx
(b)
Example 3
For the integral, (a) identify u and du and (b) integrate by substitution.
Answer
(a)
u = x + 1
du = dx
(b)
Example 4
For the integral, (a) identify u and du and (b) integrate by substitution.
Answer
(a)
u = (x2 + 3x)
du = (2x + 3)dx
(b)
Example 5
For the integral, (a) identify u and du and (b) integrate by substitution.
Answer
(a)
u = e6x
du = 6e6xdx
(b)
Example 6
Integrate by substitution.
Answer
We have
u = (5x + 7)
du = 5dx
If we break up 60 into 12 ⋅ 5, we can see how to do the substitution.
Example 7
Integrate by substitution.
Answer
We have
u = (7x2 + 11)
du = 14xdx
So
Example 8
Integrate by substitution.
Answer
We have
u = (2x2 + 4x)
du = (4x + 4)dx
It doesn't look like there's a factor of (4x + 4) in the integrand. However, (4x + 4) is the same thing as 4(x + 1). If we factor 60 into 15 ⋅ 4, we can see where the substitution happens:
Example 9
Integrate by substitution.
Answer
We choose u to be the quantity in the denominator, since we know how to integrate but we don't know how to integrate . So
u = sin (e4x)
du = 4excos(e4x)dx
Then we factor 16 into 4 ⋅ 4 and do the substitution:
Example 10
Integrate by substitution.
Answer
We have a couple of choices here. One possibility is
Then
Another possible choice is
Then
Thankfully, we get the same answer regardless of which substitution we use. We can also have integrals where a constant factor of du is missing. In such cases we can multiply by a clever form of 1 to introduce the missing factor to the integrand.
Example 11
Integrate.
Answer
Take
u = 4x2 + 6x
du = (8x + 6)dx
= 2(4x + 3)dx
The integrand is missing a factor of 2, but we can fix that.
Example 12
Integrate.
Answer
Take
The integrand is missing a factor of , so we multiply the integrand by and the integral by 2:
Example 13
Integrate.
Hint
Simplify the integrand.
Answer
This one doesn't require substitution. Following the hint, we simplify the integrand to get
We could use substitution instead, but it would take us more work to get the same answer. Take
u = 4x
du = 4dx
Then we have to multiply by a clever form of 1:
Example 14
Integrate.
Answer
Take
u = 16x2 – 10
du = 32xdx
We almost have the derivative of u in the numerator of the fraction, but we're missing the factor 32. So we multiply the integrand by 32 and the integral by .
Example 15
Integrate.
Answer
Take
The integrand is missing a factor of , but we can fix that.
but we don't know how to integrate 
. So
, so we multiply the integrand by 
and the integral by 2:
.
, but we can fix that.