Page:Calculus Made Easy.pdf/272

Table of standard forms

${\frac {dx}{dy}}$	← $y$ →	$\int ydx$
Algebraic.
$1$	$x$	${\tfrac {1}{2}}x^{2}+C$
$0$	$a$	$ax+C$
$1$	$x\pm a$	${\tfrac {1}{2}}x^{2}\pm ax+C$
$a$	$ax$	${\tfrac {1}{2}}ax^{2}+C$
$2x$	$x^{2}$	${\tfrac {1}{3}}x^{3}+C$
$nx^{n-1}$	$x^{n}$	${\frac {1}{n+1}}x^{n+1}+C$
$-x^{-2}$	$x^{-1}$	$\log _{\epsilon }{x}+C$
${\frac {du}{dx}}\pm {\frac {dv}{dx}}\pm {\frac {dw}{dx}}$	$u\pm v\pm w$	$\int udx\pm \int vdx\pm \int wdx$
$u{\frac {dv}{dx}}+v{\frac {du}{dx}}$	$uv$	No general form known
${\frac {v{\frac {du}{dx}}-u{\frac {dv}{dx}}}{v^{2}}}$	${\frac {u}{v}}$	No general form known
${\frac {du}{dx}}$	$u$	$ux-\int xdu+C$
Exponential and Logarithmic.
$\epsilon ^{x}$	$\epsilon ^{x}$	$\epsilon ^{x}+C$
$x^{-1}$	$\log _{\epsilon }x$	$x(\log _{\epsilon }x-1)+C$
$0\cdot 4343\times x^{-1}$	$\log _{10}x$	$0\cdot 4343x(\log _{\epsilon }x-1)+C$
$a^{x}\log _{\epsilon }a$	$a^{x}$	${\frac {a^{x}}{\log _{\epsilon }a}}+C$
Trigonometrical.
$\cos x$	$\sin x$	$-\cos x+C$
$-\sin x$	$\cos x$	$\sin x+C$
$\sec ^{2}x$	$\tan x$	$-\log _{\epsilon }\cos x+C$
Circular (Inverse).
${\frac {1}{\sqrt {(1-x^{2})}}}$	$\arcsin x$	$x\cdot \arcsin x+{\sqrt {1-x^{2}}}+C$
$-{\frac {1}{\sqrt {(1-x^{2})}}}$	$\arccos x$	$x\cdot \arccos x-{\sqrt {1-x^{2}}}+C$
${\frac {1}{1+x^{2}}}$	$\arctan x$	$x\cdot \arctan x-{\tfrac {1}{2}}\log _{\epsilon }(1+x^{2})+C$