Let , then
Convert to an integral in the variable : The final answer in terms of is:
In this section we learn to reverse the chain rule by making a substitution.
U-Substitution
Suppose that is an anti-derivative of , i.e., . Consider the composition, , where is a differentiable function. What is the derivative of ? According to the chain rule, From the point of view of integration, this differentiation equation is equivalent to the anti-differentiation equation: What this equation would look like if we used familar functions? We will let and . Composing them gives Furthermore, the derivative of is, , and an anti-derivative of is, . Putting this information into the anti-differentiation equation yields: This is most easily understood by differentiaing the right hand side using the chain rule(!): In other words, this equation is difficult to reproduce is we are just staring at the left hand side: The technique for systematically finding the anti-derivative, is to make a u-substitution which is a three step process. In the first step, the substitution is declared and the integral is re-written in terms of the variable as Notice that the left hand-side has the differential and the right hand side has the differential . We will learn how to convert between differentials. Step two is to compute the indefinite integral. Assuming we know an antiderivative for , we get: The third and final step is to replace the variable in the previous step with , to obtain Thus Refer back to this equation after you have worked through some examples and problems to review the method of -substitution.
example 1 Compute
We will make a u-substitution for the inside of the composition : We now determine the relationship between the differentials and . Differentiatiating gives: This equation can be written in differential form as For the purpose of our impending computation, we will express this equation in terms of the original differential, : We can now proceed with step one and convert the original integral into a simpler one, in the variable, : Notice that ’ cancelled with its reciprocal. Now for step two, anti-differentiation: And the coup de grace, step three, we return to the variable :
Recapping,
(problem 1a) Compute:
Let , then
Convert to an integral in the variable : The final answer in terms of is:
Let , then
Convert to an integral in the variable : The final answer in terms of is:
(problem 1b) Compute: .
Let , then .
Convert to an integral in the variable : The final answer in terms of is:
Let , then .
Convert to an integral in the variable : The final answer in terms of is:
example 2 Compute Let . We have,
and the integral can be written as The last integral can be computed as and by back substituting, we have Thus, using u-substitution we can conclude that
(problem 2a) Compute .
Let , then .
Convert to an integral in the variable : The final answer in terms of is:
Let , then .
Convert to an integral in the variable : The final answer in terms of is:
(problem 2b) Compute .
Let , then .
Convert to an integral in the variable : The final answer in terms of is:
Let , then .
Convert to an integral in the variable : The final answer in terms of is:
example 3 Compute Let . We have,
The last integral can be computed as and by back substituting, we have Thus, using u-substitution we can conclude that
(problem 3a) Compute .
Let , then .
Convert to an integral in the variable : The final answer in terms of is:
Let , then .
Convert to an integral in the variable : The final answer in terms of is:
(problem 3b) Compute .
Let , then .
Convert to an integral in the variable : The final answer in terms of is:
Let , then .
Convert to an integral in the variable : The final answer in terms of is:
example 4 Compute Let . We have,
The last integral can be computed as and by back substituting, we have Thus, using u-substitution we can conclude that
example 5 Compute Let . We have,
The last integral can be computed as and by back substituting, we have Thus, using u-substitution we can conclude that
example 6 Compute We let . We have,
and the integral can be written as The last integral can be computed as and by back substituting, we have Thus, using u-substitution we can conclude that
example 7 Compute Let . Then we have,
and the integral can be written as
The last integral can be computed as and by back substituting, we have Thus, using u-substitution we can conclude thatexample 8 Compute Let . Then,
and the integral can be written as
The last integral can be computed as and by back substituting, we have Thus, using u-substitution we can conclude thatexample 9 Compute Let . Then we have, and the integral can be written as
The last integral can be computed as and by back substituting, we have Thus, using
u-substitution we can conclude that
example 10 Compute Let . Then we have, and the integral can be written as The
last integral can be computed as and by back substituting, we have Thus, using
u-substitution we can conclude that
example 11 Compute Let . Then we have, and the integral can be written as The
last integral can be computed as and by back substituting, we have Thus, using
u-substitution we can conclude that
example 12 Compute Let . Then we have, and the integral can be written as The
last integral can be computed as and by back substituting, we have Thus, using
u-substitution we can conclude that
example 13 Compute Let . Then we have, and the integral can be written as The
last integral can be computed as and by back substituting, we have Thus, using
u-substitution we can conclude that
example 14 Compute Let . Then we have, and the integral can be written as The
last integral can be computed as and by back substituting, we have Thus, using
u-substitution we can conclude that
example 15 Compute First rewrite the integral: Now, let ; then and the integral can be written
as
The last integral can be computed as and by back substituting, we have Thus, using
u-substitution we can conclude that
example 16 Compute First rewrite the integral: Distributing in the numerator, we
get Now, let , then and the integral can be rewritten as The last integral can be
computed as and by back substituting, we have Thus, using u-substitution we can
conclude that
Here is a detailed, lecture style video on u-substitution:
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