626 Molesworth’s pocket-book
Differential and Integral Calculus—continued.
Radical expressions must be represented by fractional indices, thus
\Tx'i must be expressed as x3 ; —7= as x~^; sin.-1 x, cos.-1 x, &c.,
NjX
denote the arc whose sine or cosine, &c., is x.
Differential Coefficients.
then — — j and du — Integrals.
Ifu — dx \
a. 0. 0.
X 1 dx /dx = x + C
nx n ndx f ndx = nx + C
XH nxn ~ 1 nxn 1 dx f nx" “1 dx xn + C
X2 2x 2x.dx ,/'2ax.dx = ax2 + C
X3 3æ2 3x2 dx f 3ax*.dx = ax3 + C
— X - '2 ^dx ax ~ ■’ dx = - ax " i + C
1 =ZX 3 — X ” 3 — 2.x ~ 1 dx f- fax - 1dx = ax“l+C
æf axs + bx+c 2aæ + b (2ax + 6) dx /(2ax+b) dx—ax2+hx+C
log. b x *111 X m , — dx X / ” dx = log. x + C J X 6
■[log. t x 1 dx ,• dx / — = log. e x + C
X X </ X
ax a? log. e a aj: log. f adx f ax log-. e adx - ax + 0
sin. x COS. X cos. xdx f cos. xdx = + sin. x + C
cos. x — sin. x — sin. xdx J sin. xdx = — cos. x + C
tan. x 1 dx
COS.2 X cos.2 X J COS.2X
cot. X 1 dx /• dx - cot. X + C
* si«.2 X sin.2 x • J sin.2 x
see, x sin. x sin. xdx /• sin. xdx , , — „ = + sec. x + C
cos. 2 X cos.2 X J cos. zx
cos. X cos. x.dx /’cos. xdx _ , ~ /——-— =-cosec. x+C «/ sin.2 x
cosec. x
sin.2 x sin.2 x
* m is the modulus of the system of logarithms haring a base b.
r e rr. the number whose hyp. log. is 1 = 2'7182816.