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Substitution

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    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

          \[ \int (f(g(x))' dx = \int(f'(g(x))g'(x) dx \]

     


    Examples

    Example

    Calculate

            \( \int \frac{2x}{(x^2 + 1)^2 } dx \)

     

    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

            \( = \frac{1}{202} (x^2 + 4)^{101} + C \)
     

    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.   \( \int (x^2 + 1) \sqrt{x^3 + 3x} dx \)        

     

    Back to the Integration and Applications Page

     
     

    Substitution is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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