Calculus Made Easy/Table of Standard Forms

by Silvanus Phillips Thompson

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).
${\frac {1}{\sqrt {(1-x^{2})}}}$	$\arcsin x$	$x\cdot \arcsin x+{\sqrt {1-x^{2}}}+C$
$-{\frac {1}{\sqrt {(1-x^{2})}}}$	$\arccos x$	$x\cdot \arccos x-{\sqrt {1-x^{2}}}+C$
${\frac {1}{1+x^{2}}}$	$\arctan x$	$x\cdot \arctan x-{\tfrac {1}{2}}\log _{\epsilon }(1+x^{2})+C$
Hyperbolic.
$\cosh x$	$\sinh x$	$\cosh x+C$
$\sinh x$	$\cosh x$	$\sinh x+C$
$\operatorname {sech} ^{2}x$	$\tanh x$	$\log _{\epsilon }\cosh x+C$
Miscellaneous.
$-{\frac {1}{(x+a)^{2}}}$	${\frac {1}{x+a}}$	$\log _{\epsilon }(x+a)+C$
$-{\frac {x}{(a^{2}+x^{2})^{\tfrac {3}{2}}}}$	${\frac {1}{\sqrt {a^{2}+x^{2}}}}$	$\log _{\epsilon }(x+{\sqrt {a^{2}+x^{2}}})+C$
$\mp {\frac {b}{(a\pm bx)^{2}}}$	${\frac {1}{a\pm bx}}$	$\pm {\frac {1}{b}}\log _{\epsilon }(a\pm bx)+C$
${\frac {-3a^{x}x}{(a^{2}+x^{2})^{\tfrac {5}{2}}}}$	${\frac {a^{2}}{(a^{2}+x^{2})^{\tfrac {3}{2}}}}$	${\frac {x}{\sqrt {a^{2}+x^{2}}}}+C$
$a\cdot \cos ax$	$\sin ax$	${\frac {1}{a}}\cos ax+C$
$-a\cdot \sin ax$	$\cos ax$	${\frac {1}{a}}\sin ax+C$
$a\cdot \sec ^{2}ax$	$\tan ax$	$-{\frac {1}{a}}\log _{\epsilon }\cos ax+C$
$\sin 2x$	$\sin ^{2}x$	${\frac {x}{2}}-{\frac {\sin 2x}{4}}+C$
$-\sin 2x$	$\cos ^{2}x$	${\frac {x}{2}}+{\frac {\sin 2x}{4}}+C$
$n\cdot \sin ^{n-1}x\cdot \cos x$	$\sin ^{n}x$	${\frac {\cos x}{n}}\sin ^{n-1}x+{\frac {n-1}{n}}\int \sin ^{n-2}xdx+C$
$-{\frac {\cos x}{\sin ^{2}x}}$	${\frac {1}{\sin x}}$	$\log _{\epsilon }\tan {\frac {x}{2}}+C$
$-{\frac {\sin 2x}{\sin ^{4}x}}$	${\frac {1}{\sin ^{2}x}}$	$-\operatorname {cotan} x+C$
${\frac {\sin ^{2}x-\cos ^{2}x}{\sin ^{2}x\cdot \cos ^{2}x}}$	${\frac {1}{\sin x\cdot \cos x}}$	$\log _{\epsilon }\tan x+C$
$n\cdot \sin mx\cdot \cos nx+m\cdot \sin nx\cdot \cos mx$	$\sin mx\cdot \sin nx$	${\tfrac {1}{2}}\cos(m-n)x-{\tfrac {1}{2}}\cos(m+n)x+C$
$2a\cdot \sin 2ax$	$\sin ^{2}ax$	${\frac {x}{2}}-{\frac {\sin 2ax}{4a}}+C$
$-2a\cdot \sin 2ax$	$\cos ^{2}ax$	${\frac {x}{2}}+{\frac {\sin 2ax}{4a}}+C$

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