188 PRACTICAL MATHEMATICS 



1 7 70 



Solving for Y, Y = - ef* cos to -\ tf sin bx -- 3 Y 

 a a 2 a 2 



Y(a 2 + b 2 ) = ef KK (a cos to + b sin to) 



g 



Y = -5 TO (a cos bx+ b sin 

 a^ + er v 



f e?* b 



or I eF 1 cos bx dx = , cos (to a) where tan a = - 



108. For the integration of e? x sin w x or e^ cos 71 x, sin" x and 

 cos n x must be expressed in terms of the sines or cosines of the 

 multiple angles of x. Then each of the integrals will split up 



into a number of integrals each of the form I e sin to dx or 



Y?* cos to dx, and these can be integrated by the previous 

 method. In this case though, because it will be necessary to 

 work with I ef sin to dx and le * cos to dx for various 



numerical values of b, it is best to establish the results working 

 with a and b, and use them as standard forms. 



(a) To integrate e** sin 5 x. 



f 1\ 5 



Now (2i sin a;) 5 = f y j , if y = cos x + i sin x 



32i sin 5 x = y 5 5/ 3 + 10?/ 10- + 5-= = 



?/ tl ?/ 



-D +: 



= 2i sin 5a; 10 sin 3a; + 20i sin x 



155 



and sin 5 x = sin 5a; sin 3a? 4- - sin x 

 ID ID o 



Then I e** sin 5 x dx = e** sin 5a; da; I e 2x sin 3# da; 

 J 1G J 16 J 



5f 

 + - \e 2x sin a; da; and each of these integrals can be determined 



8 J 



by using as a standard form 



f (P^ b 



I e?* sin to da; = . sin (to a) , where tan a = - 



J A/a 2 + b 2 a 



Hence I e^ sin 5 x dx = 7= sin (5# a) 7= sin (3a? (3) 



16A/29 16V13 



fj O 1 



+ -^-p= sin (a; y), where tan a = -, tan [i = - and tan y = - 



