Integration using Euler's formula

<p>In <a href="/facts/Integral_calculus/V7lyV997">integral calculus</a>, <a href="/facts/Euler%2527s_formula/Wq7VXIV0">Euler's formula</a> for <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> may be used to evaluate <a href="/facts/Integral/V7lyV997">integrals</a> involving <a href="/facts/Trigonometric_functions/czej7ur0">trigonometric functions</a>. Using Euler's formula, any trigonometric function may be written in terms of complex exponential functions, namely 
  
    
      
        
          e
          
            i
            x
          
        
      
    
    {\displaystyle e^{ix}}
  
 and 
  
    
      
        
          e
          
            −
            i
            x
          
        
      
    
    {\displaystyle e^{-ix}}
  
 and then integrated. This technique is often simpler and faster than using <a href="/facts/Trigonometric_identities/Xw9Jx8Vz">trigonometric identities</a> or <a href="/facts/Integration_by_parts/aV6JgBNd">integration by parts</a>, and is sufficiently powerful to integrate any <a href="/facts/Rational_fraction/rgzXYjyr">rational expression</a> involving trigonometric functions.
</p>

Integration using Euler's formula open-in-new

Integration using Euler's formula