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5.5 – The Substitution 5.5 – The Substitution RuleRule

2

Example – Optional for Pattern Learners

1. Evaluate 23 6 .xe x dx

3. Evaluate 2cos tan sec .x x dx

Use WolframAlpha to evaluate the following.

2. Evaluate 3sin 4 2 .x x dx

Notice that each of these are of the form where u is some function of x. If is the antiderivative of f is F, what will be the answer of an indefinite integral of this form?

,f u u dx

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The Substitution Rule – The Idea

sin cosxe x dxEvaluate

Clearly, the answer is e sin x + c. How do we arrive at this?

Let u = sin x. Then du/dx = cos x. Substituting this into the integral above, we get

u udue dx e du

dx

This is a simple integral that we can evaluate and get eu + c. Substituting sin x in for u, we have our answer.

\\

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The Substitution Rule

If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then

f g x g x dx

Since,

u g x

dug x

dxdu g x dx

u du

f u du

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The Substitution Rule – General Technique

1. Let u be g(x) in the composition. You may wish to rearrange the function so that anything not in the composition is in front of the dx.

2. Determine du/dx and multiply both sides by dx.

3. Divide both sides by the constant, if necessary.

Note: There are variations on this technique so it may have subtle changes in process. Also, you may have a different variable than x so us du/d[variable] in step 2.

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Examples - Evaluate

92 3

22

3 4

1

2

1. 3 5

2.1

3. 2 1

4. sec 2 tan 2

tan5.

1

x x dx

xdx

x

y y dy

d

xdx

x

3 / 2

5 2

cos

2

6. sin 1

7. 1 tan sec

8. sin

sin9.

1 cos

10. tan

t

x x dx

d

e t dt

xdx

x

x dx

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Substitution Rule Twists - ExamplesSometimes things are not as obvious and it may not seem that you can find a u = g(x) such that du = g'(x) dx. With a little creativity, you can. You may need to solve for x in terms of u. Evaluate the following.

23 3 51. 1 2.

1

xx x dx dx

x

This uses the standard substitution technique, but has a little twist.

43.

1

xdx

x

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The Substitution Rule – Definite Integrals

If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then

( )

( )

b g b

a g af g x g x dx f u du

Properties: Suppose f is continuous on [– a, a].

0

( ) If is even , then ( ) 2 ( ) .

( ) If is odd , then ( ) 0.

a a

a

a

a

a f f x f x f x dx f x dx

b f f x f x f x dx

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Examples - Evaluate

2

2

0

1

0

2/ 2

6/ 2

4

0

11/ 2

0 2

1. cos

2.

sin3.

1

4.1 2

sin5.

1

x

x x dx

xe dx

x xdx

xx

dxx

xdx

x