Substitution

Substitution (Definition)

Recall that the chain rule states that

[f(g(x))]'   =   f '[g(x)] g'(x)

For example,

(1 - x2)7 '  =  7(1 - x2)6 (-2x)

The goal of this discussion is to learn how to go backwards.  This process is called integration by substitution.  In general we have

Examples

Example

Calculate

Solution

Let

u =  x2 +1

du/dx  =  2x     or     du  =  2x dx

We substitute:

u -2 du  =  -u -1 + C  =  -(x2 + 1) -1 + C

Steps in Substitution

1. Find the function derivative pair (f and f ').

2. Let u = f(x).

3. Find du/dx and adjust for constants.

4. Substitute.

5. Integrate.

6. Resubstitute.

Example

Integrate

x (x2 + 4)100 dx

Solution

The difficulty with this is that the derivative of x2 + 4 is 2x not x.  We can overcome this by multiplying and dividing by 2

x (x2 + 4)100 dx  = 1/2 2x (x2 + 4)100 dx

Now we can use substitution.  Let

u  =  x2 + 4        du  =  2x dx

The integral becomes

1                         1     1
u100 du  =                u101 + C
2                         2   101

1
=           (x2 + 4)101 + C
202

Example

Integrate

(x2 + 2)2 dx

Solution

The difficulty with this is that the derivative of x2 + 2 is 2x.  There is more than a constant that we are lacking.  The method of substitution will not work here.  Instead, we use some algebra (FOIL).

(x2 + 2)2 dx  =   (x4 + 4x2 + 4) dx

Now we can use the power rule to get

=  1/5 x5 + 4/3 x3 + 4x + C

Exercises

Integrate

A. x2 (x3 + 6) -5 dx

B.

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