TABLES 9-15.— SOME MATHEMATICAL TABLES 



23 



TABLE 9.— DERIVATIVES AND INTEGRALS 



, unless » = — 1 



.// N j f(>() du 



d f ( u) =d —z — • -r- ■ d x 



an ai- 

 de* = e* dx 

 d e- = a c" dx 



dlog.x = — dx 

 x 



dx r = x* (1 -flog, x) dx 



dsinx — cos x dx 

 d cos .r =r — sin .r dx 



d tan 



— sec 2 .r dx 



d cot x = — esc 2 x dx 



d sec x = tan x sec x dx 



d esc .r = — cot x ■ esc x dx 



rfsiiT\r = (1 — x*y*dx 



dcos'Vr =- (1 -x 2 )-*dx 



dtan'x = (1 + x 2 )- 1 dx 



dcot^x =— (1 + x*y i dx 



dsec^x = x~ 1 (.r 2 — l)-» dx 



dcsc~ x x = — x- 1 (x*—l)-*dx 



dsinhx = coshxdx 



d cosh x = sinh x dx 



d tanh x = sech 2 x dx 



d coth x = — csch 2 x dx 



d sech .r = — sech x tanh .r dx 



d csch x = — csch x-coth x dx 



dsinh" 1 x = (x 2 -f l)"»dx 



dcosh' 1 x= (x 2 — l)" J rf.r 

 dtanh^x^ (1— x*Y x dx 

 rfcath-*jr = (1 — x 2 )" a dx 

 d sech" 1 x = — x" 1 ( 1 — x 2 ) J dx 

 d csch' 1 .v = — .r" 1 (x 2 -f 1)"' dx 



-i 



'dx 



J (a 2 + x 2 rdx 

 Ka'-x'yVx 



/(a'-x 2 )- } dx 



/x(a' ± x*y*dx 

 /sin 2 .r dx 

 /cos 2 x dx 

 /sin x cos x dx 



1 



= — tan 

 a 



1 



snr 



X 



Vx 2 + a* 



, a + x 



-r- log 



la a — x 



= sin — , or — cos — 

 a a 



= ±(a 2 ±x 2 )* 



= — i cos x sin x + i-*" 



= \ sin x cos x + -Jx 



= \ sin 2 x 



/(sin x cos x)~'dx = log tan x 



/tan x dx 



/tan 2 x dx 



/cot x dx 



/cot 2 x dx 



/esc x dx 



/x sin x dx 



/x cos x dx 



/tanh x dx 



/coth x dx 



/sech x dx 



/csch x dx 



fx sinh x dx 

 fx cosh x dx 

 /sinh 2 x dx 

 fcosh 2 x dx 



= — log cos x 

 = tan x — x 

 = log sin x 

 = — cot x — x 

 = log tan ix 

 = sin x — x cos x 

 = cos x + x sin x 

 = log cosh x 

 = log sinh x 

 = 2 tan~V = gd u 



= log tanh - 



= r cosh x — sinh x 

 = x sinh x — cosh x 

 = t (sinh x cosh x — x) 

 = V (sinh x cosh x -f x) 



/sinh x cosh x dx = 1 cosh (2x) 



SMITHSONIAN PHYSICAL TABLES 



