DIFFERENTIAL AND INTEGRAL CALCULUS. 25 



(Such however had better be done at once by assuming 

 A B 



x* -f 4:X + 3 x + 1 x + 3 

 _ 

 ~ 



(g N / I 1 



^)=-2V(3-2aj-^)+sin- 1 (- - 

 / \ J 



; 



/y, I 1 



tan . 



The other general method employed for effecting in- 

 tegrations is what is known as " integration 'by parts," 



and is founded on the formula ^ (uv) = u- r + v- T -. whence 



ax ^ dx ax 7 



du , c/v dv .. du 



uv 



/ du , f c/v [ dv .. [ du 7 



\v T-dx+ \u -7- J?, or lu -r-dx=uv \v r dx. 

 J dx J dx J dx J dx 



Thus, to integrate cc sinaa;, put w = #, -7- = sin ace, and 



ttC 



therefore v -- cosaa;, in the formula, when 



/ . x [cosax , x smax 



lx smaxdx cosacc+ dx = cosax-\ --- 5 : 

 J a J a a a 



rifi) 



similarly for x 2 cosao;, put u x z ^ -j- cos ax, and therefore 



v = - sinaic, whence 



[ * a? 2 f 



ar cosacc6/a: = sinaa? --- lx s 

 J a a] 







 smaxdx 



x z . 2x 2 



= sin ax H 5- cosao; -- r. 

 a a 2 a 



