26 DIFFERENTIAL AND INTEGRAL CALCULUS. 



Again, to integrate e mx sin nx } mx dv 



} , put (1) M=X l * T - = si 

 and e cos me j 7 ' dx 



therefore v = cos wo?, or 

 n 



f mx j mx cosnx m f mx 



le smnxdx = e - -\ le cosnxdx ..... (A) ; 



and (2) put u = sinnx, -r- = e mx . and therefore v = e 

 1 dx in 



and we have 



/>* smnxdx = e mx ^^ - - (e mx cosnxdx ..... (J5). 

 J mm) 



(A) and (5) together determine both integrals, subtract 



(B) from (-4), and we get 



fm n\ [ )nx , , nr (cosnx sn??a?\ 



- + e cosnxdx = e - + 



\n mj J \ n m J 1 



f m * x (m co&nx + n smnx] 



or e coswa; dx = e * -, - ^ - , 



J m + n 



multiply (A) by , (B) by and add, when we get 



(m n\ [ mr . , m , fslnnx cosnx\ 



- + - smnxdx = e : 

 \ n mj J \ n m J 



E :"'* a 



amnxdx=e 



nr (m smnx n cosnx] 

 - 



m' + n 2 



Another important integral JV(a 2 x*) dx may be found 

 by putting u = \/(a" as*), -j- = 1, and therefore v = #, then 



and 



ar^-J-^fes? fcr*-*;-/Vfc*-rf)*ri 



