12 



Table 5. 

 DIFFERENTIAL COEFFICIENTS. 



INTEGRALS. 



DIFFERENTIAL COEFFICIENTS. 



«=«" 



loge« 



sin. X 



COS. X 



tan. X 

 cot. X 



sec. X 

 cosec. X 



ax 



tan.—* X 



cot.- 



cosec.—' 3f 



covers.—* x 



a^ loge a 

 a; 



COS. « 



—sin. X 

 sec' :» 

 —cosec' X 

 sin. Jg 

 COS.' ;c 



COS.* 



V(i-^') 

 I 



I 



i+x"^ 

 I 



I 



x\/{x'^—i) 

 I 



5C\/(»'— l) 



I 



v/lz x—x^) 



I 



■v/(2 a;— *2) 



INTEGRALS. 



fx^dx 



fa^dx 



SeUx 

 fdx 



*f X 



y*cos. ax • dx 



ysin. ax • dx 



/sec' ax • dx 



/cosec' ax- dx 

 sin. 5c^ 



/ 



/ 



COS.* X 



cos, a; 

 sin.' X 



dx 



dx 

 dx 



\/{a^-x^) 



r dx 



Ja'+x' 



r dx 



J x^(x^-a^) 



A 



dx 

 dx 



\/(2X—X^) 



n+i 

 a' 



loge* 

 loge» 



sin, ax 



a 

 —cos. ax 



a 

 tan, ax 



a 

 —cot, ax 



a 

 sec. X 



—cosec. X 



f • _i ^ 



' sin. ' - 



a 



— COS.— '- 



a 



' I . , a; 

 - tan.-' - 

 a a 



cot.-' - 



I , X 



- sec.—' - 

 a a 



cosec ' — 



a a 



vers.—' X 

 —covers.—' x 



Taylor's series : 



u=f{x+h)=f{x) +f'(x)h+r(x) ^ +f'"(x) — ^ +• 



2 I • 2 • 3 



The remainder after the first n terms is expressed by 



■X"/"+'(*+A-s)s"'<^z- 



I .2 -3 ...n 



Maclaurin's series : 



x' 



u=J(x)=J{o) +f'(o)x+J"(o) -^ +r'(o) -—- + • 



I •2 I • 2 • 3 



^=3.14159265359 

 1=0.31830988618 



7r'=9.8696o440i09 



6=2.71828182846 



v't=i.7724S38509I 



^=0.88622692546 



log,oir=o.497i4987269 

 logioe=o.43429448i90 

 loge 10=2.30258509299 

 loga(number) =loge (number) • log^ e 

 __logp(number) 

 log,B 



Smithsonian Tables^ 



