Substitution 

Substitution

Recall that the chain rule states that

        [f(g(x))]'  =  f '[g(x)] g'(x)

Integrating both sides we get:


Example

Calculate

int(2x/(x^2+1)^2)dx = int(2x(x^2+1)^-2)dx

Let 

        u =  x2 +1

        du/dx = 2x     or     du = 2xdx

We substitute:

       


Steps

  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



Exercises

Integrate the following:

       

and

        

 

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