page_title: Trigonometric substitution
section_title: Examples of Case III
hierarchical_section_title: Trigonometric substitution / Case III: Integrands containing 
    
      
        
          
            x
            
              2
            
          
          −
          
            a
            
              2
            
          
        
      
      {\displaystyle x^{2}-a^{2}} / Examples of Case III
caption_attribution_description: English: Geometric construction for the trigonometric substitution x = a sec &#x2061; &#x03B8; {\displaystyle x=a\sec \theta } .
context_page_description: In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. In calculus, trigonometric substitution is a technique for evaluating integrals. Moreover, one may use the trigonometric identities to simplify certain integrals containing radical expressions. Like other methods of integration by substitution, when evaluating a definite integral, it may be simpler to completely deduce the antiderivative before applying the boundaries of integration.
context_section_description: Integrals like
  can also be evaluated by partial fractions rather than trigonometric substitutions. However, the integral
  cannot.  In this case, an appropriate substitution is:
  where  so that , and  by assuming , so that  and .
  Then,
  One may evaluate the integral of the secant function by multiplying the numerator and denominator by  and the integral of secant cubed by parts. As a result,
  When , which happens when  given the range of arcsecant, , meaning  instead in that case.
caption: Geometric construction for Case III
exif: {}