More Integrals involving Trigonometric Functions

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Trig

 

In a Nut Shell:  General guidelines for integrals involving products of tangent and of

secant depend on whether the powers of tangent and secant are even or odd.  Here is

a guideline.

 

With m an odd integer and n an even integer;  i.e.

 

Example 3:      I1  =   ∫ tanm x secn x dx  

 

 

 

I1  =   ∫ tan3 x sec2 x dx                  Now  let  u  = tan x   du =  sec2x dx

 

   So   I1   =    ∫ u3  du     which is a standard integral.

 

Note:  Similar reasoning for the integral:   I1a  = ∫ cotm x cscn x dx  

 

 

With m an even integer and n an odd integer;  i.e.

 

Example 4:    (much harder)  I2  =   ∫ tanm x secn x dx  

 

 

        I2  =   ∫ tan2 x  sec3 x dx  

 

      tan2x =  sec2 x  -  1

 

        I2  =   ∫ (1 + sec2 x) sec3 x dx  

 

        I2  =   ∫ (sec5 x - sec3 x)dx     Next use integration by parts

 

        on each of these 2 integrals;  first,  I2a  =   ∫sec3 x dx  

 

  Next let    u  =  sec x                   dv =  sec2x dx

                 du  =  sec x tan x dx      v  =  tan x

 

 I2a  =   ∫sec3 x dx   =  sec x tan x - ∫sec x tan2 x dx

 

Click here to continue with Example 4.

 

 Copyright © 2011 Richard C. Coddington

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