INTEGRATION BY PARTS 189 



sin' . dr = _ sin5*-tan-'-..sin (te 



(6) To integrate e 3 * cos 4 a?. 



Now (2 cos a?) 4 = (y + -Y, if t/ = cos x + i sin * 



16 cos 4 a: = t/ 4 + 4t/ 2 + 6 + 4 + -4 



= 2 cos 4a; + 8 cos 2x -f 6 



113 



cos 4 x = - cos 4a; + - cos 2x 4- - 

 828 



Then t e 3 * cos 4 x dx = - \e^ cos 4 x dx -f- - I &* cos 2a; (ic 



3 f f *** 



+ - le 3 * dx and using I e * cos bx dx = 7 ==. cos (bx a 

 J V a 2 + b 2 



where tan a = -, as a standard form. 

 a 



f 



le 



g 



cos 4 x dx = cos (4 - a) H -- -=* cos (2 - p) + 



where . tan a = ^ and tan Q = - 



o o 



Then 



P e** fl / 4\ 



I e 3 * cos 4 x dx = |- cos f^c-tan- 1 -) 



mm cos (tat - tan- 1 - ) + 1 } 

 l3 3 / 



*-m 

 Vl3 



(c) To integrate e* 8 sin 2 x cos 3 a?. 



/ IN 2 / 1\ 3 



Now (2i sin #) 2 (2 cos a:) 3 = (y - -) (y + - J , if y = cos x + i sin x 



\ t/ \ y / 



- 32 sin 2 cos 3 x=(y z - -jj ^ + -) 



= 2 cos 5x + 2 cos 3a; 4 cos x 



sin 2 a; cos 3 x = - cos x - cos 3a; - cos 5a; 

 8 16 lo 



