DIFFERENTIAL AND INTEGRAL CALCULUS. 37 



4. If x =/(z), prove that /< (x) dx =/< {/(*)} ./ (z) dz ; 



. C dx .. C dx 



and obtain the integral* , and 



by putting (1) x a (1 coss), (2) # = sees. 

 5. Prove the formula for integration by parts, 



C dv , [ du j 



\ u -j- ax = wv / v -y- 05. 

 j dx ] dx 



Apply it to find the integrals of a; cosa#, x z sinaa?, cc n logic, 

 e'" x cosrc#, and cos ax cos5a?. 



6. Shew how to integrate 



1 1 , 1 



' 



a -f foe 4- ca; 2 ' (a? + a) 



7. Integrate 



1 1 1 



a 4- b cosx ' a + b smx ' a + b cosx + c sina; ' 

 (when a > b in the first two and a 2 > V* -f c 2 in the third), 



also integrate -- . -s , -. ; or deduce them 

 cosa; smic cosic-fsmic 



from the former. 



8. Prove that a 2 - a; 2 dx 



[ ^(a 2 - a; 2 ) 



J 



= 2 ( *J (2ax - a?) dx, 



. J 



and I x *J(2ax x 2 ) dx = %7ra 3 . 



J o 



r 1 i 



9. Prove that I x r logic dx=- - -^ , 



f *" (lo g;E ) a fa = 4 ; f aT 1 



J n ' J e\ 







and deduce 



