Xxiv USEFUL FORMULAS. 



Since d{uv) = icdv ■\- vdu, 



I udv = iw — i vdu ; and 



if u := f{x) and Z' = <f> (x), 



fdxj/(x, J') dy =ydjj/(x, y) dx. 

 I dx \f{x) dx = X i/(x) dx — I x/{x) dx. 



Cx''dx = -4— X"-' 4- C. 

 J n-\-i ' 



/dx I , , 



— = log ^ t + C-, j ^j^p^ == ^- Mog (« + ^^). 



I 7ir-p = arc tan x -^ C, J , .^2 = arc cot ;c + C. 



/dx , I H- :v , ^ P dx , X — I , ^ 



— - , — -J— 2 = (^^')~^ arc tan (^/«)^ ^t -|- C, for a and Z- both positive, 



= (ab)~^ arc cot (/'/^)^ ^ "h ^> f<^^ <^^ and b both negative, 



( — aby^ — bx 

 = y,{-ab)-'- log ^_ ^^|i _^ /,_^ + C, for t?^ negative. 



/dx b -\~ ex 



a+2bx + cx' = (^^ - ^")~' ''^'^ ''''' (:^F^^y + ^' ^°'" ""' - ^" > °' 



(^^ d^^ b ex 



=^ \ (b^ — a/:)~i log Vp -1 I , I + C, for b"^— ac^ o. 



2 ^ ' ^ {Ir — acy -\-b -\-cx ' ' 



C{a + ;c'^)5 dx=\x{a^ x-f -^ }, a log {a; + {a + a:'^)*} + C. 

 I (a*^ — jr^)* dx = I X (a^ — x^y -\- \ (i^ arc sin - -|- C. 



C(a + bx)i dx = ?j (a + bxy/b + C 



* This is the formula for integration by parts. 



t Natural logarithms are used in this and the following integrals. For relation of natural to 

 common logarithms see section i, g. 



