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.