Trigonometric Substitution

1. Integration Involving Sqrt(a² - x²) 
   When we have integrals that involve the above square root, we can use the appropriate substitution.

   Example:

   

   We have

   

   The class will try the following:

   1. int x/sqrt(1 - x²) dx

   2. int sqrt(1 - x²)/x⁴dx

   3. int 1/sqrt(4-9x²⁾dx 
      

2. Two key formulas

   From Trigonometry, we have the following two key formulas:
   

   A)  sec2 x = 1 + tan2 x  so that sec x = sqrt(1 + tan2 x) B)  tan2 x = sec2 x - 1 so that tan x = sqrt(sec2 x - 1)

   When we have integrals that involve any of the above square roots, we can use the appropriate substitution.

   Example:

   

   The class will try the following:

   1. int x³/sqrt(x² - 1) dx 

   2. int x²/sqrt(9 +4x²) dx

   3. int 1/sqrt(x² + 2x) dx