USEFUL FORMULAS. XXlll 



Since x is the least value of (x -\- /i — z) within the limits of this integral, the 

 sum of the remaining terms is negative, and numerically 



<J''(^)'- 



If, for example, (///.r) ^= i/ioo, the remainder in question is less than 

 \ X 0.434 X Io~^ or about one unit in the ninth place of decimals. 



d. Example of Maclaurin's series. 



u =/(x) :=■ sin X. 



f{o)=o. 



Hence 



/ (x) ^ sm X ^ X — — :; — : — z 



J ^ ' I. 2.3'!. 2. 3. 4. 5 



and the sum of the remaining terms is 



X 



/sin {x — z) z^ dz. 



5 

 o 



If g is the greatest value of sin (x — z) within the limits of this integral the 

 remainder in question is negative and numerically 



If, for example, x = ir/6 (the arc of 30°), g= h and the remainder is numeri- 

 cally less than 0.0000143. 



7. Elementary Formulas for Integration. 



a. Indefinite integrals. 



I adx z=. a \ dx =. ax -\- C. 



jfix) dx + JcA {x) dx = f{/(x) + <^ (x)} dx. 



If jc = c/) (j), and dx = ^' (j>) dy, 



f/(x) dx =f/ {cf> ( J')} ^' (y) dy. 



d_ 

 dy 



f/(x,y)dx = f^dx. 



