USEFUL FORMULAS. XXV 



C(a + 2 i^x 4- c-xy dx = I (/^ + ex) (a + 2 />x + cxj/c 



+ K^^ - '^')/'-/('^ + - ^'^"^ + ^^')"* ^^-^" + ^■ 

 Ca + ^a-)-i ./a- =2(a-\- bxflb + C 



r(a -f- ^.v) (<? + /u-)-* dx = 2. (3 a^ - 2 flr/3 + /3 ^.v) {a + ^x)V^' + C. 



r(^2 _ ^.2)-i ^_^. _ i arc sin '^ + C, 



= ^ arc cos ^ -|- C, 

 = 2 arc tan ^^^^-^-^J +C 

 r(a + a-2)-i ^x = log {x + (« + a-^*} + C, 



C{a + 2 bx + ^a-^r* ^a- = ^ log {^ + ^-v + 0?r + /^^A- + rA--)!}+C,forr>o, 



I log a: ^JC = X log a: — a- -f- C. 

 J(log .r)" x-^ dx = -j^^ (log x)" + 1 + C. 

 Tsin x dx = — cos x -\- C, J cos x dx = sin .r + C 



Tsin^ xdx = ^x — Isin 2 x-\-C, J cos^ :c //x = ^ a: + 1 sin 2 x+ C. 



ftan .r dx = — log cos x -\- C, j cot a: ^/x r= log sin x -^ C. 



J-^. = log tan i .v+ C, J^-^ = logtan i (x + h -) + ^- 



/ . f ■ = — cot X -\- C, I — •>— = tan X 4- C. 



sin^ a; ^ J cos- x ' 



Jrt! sin bx — b cos ^x „^ , _ 

 e^ sin ^.v ^x = ^^2 I ^,2 ^ + <^- 



r ax / V ^cos/^j:4-/J^sinZ'.v 



I ^"^ cos bx dx = ^2 i ^2 ^ -j- C. 



I arc sin a- ^/Iv = -v arc sin .v ± (i — x^ + C. 



I arc cos x dx = x arc cos a- =F (r — x^ -\- C. 

 I arc tan x dx = x arc tan x — i log (i + .v") + <^- 

 I arc cot X dx = x arc cot x + |- log (i + .v") + C*. 



