# Calculus Made Easy/Table of Standard Forms

Table of standard forms

${\displaystyle {\frac {dx}{dy}}}$ ${\displaystyle y}$ ${\displaystyle \int ydx}$
Algebraic.
${\displaystyle 1}$ ${\displaystyle x}$ ${\displaystyle {\tfrac {1}{2}}x^{2}+C}$
${\displaystyle 0}$ ${\displaystyle a}$ ${\displaystyle ax+C}$
${\displaystyle 1}$ ${\displaystyle x\pm a}$ ${\displaystyle {\tfrac {1}{2}}x^{2}\pm ax+C}$
${\displaystyle a}$ ${\displaystyle ax}$ ${\displaystyle {\tfrac {1}{2}}ax^{2}+C}$
${\displaystyle 2x}$ ${\displaystyle x^{2}}$ ${\displaystyle {\tfrac {1}{3}}x^{3}+C}$
${\displaystyle nx^{n-1}}$ ${\displaystyle x^{n}}$ ${\displaystyle {\frac {1}{n+1}}x^{n+1}+C}$
${\displaystyle -x^{-2}}$ ${\displaystyle x^{-1}}$ ${\displaystyle \log _{\epsilon }{x}+C}$
${\displaystyle {\frac {du}{dx}}\pm {\frac {dv}{dx}}\pm {\frac {dw}{dx}}}$ ${\displaystyle u\pm v\pm w}$ ${\displaystyle \int udx\pm \int vdx\pm \int wdx}$
${\displaystyle u{\frac {dv}{dx}}+v{\frac {du}{dx}}}$ ${\displaystyle uv}$ No general form known
${\displaystyle {\frac {v{\frac {du}{dx}}-u{\frac {dv}{dx}}}{v^{2}}}}$ ${\displaystyle {\frac {u}{v}}}$ No general form known
${\displaystyle {\frac {du}{dx}}}$ ${\displaystyle u}$ ${\displaystyle ux-\int xdu+C}$
Exponential and Logarithmic.
${\displaystyle \epsilon ^{x}}$ ${\displaystyle \epsilon ^{x}}$ ${\displaystyle \epsilon ^{x}+C}$
${\displaystyle x^{-1}}$ ${\displaystyle \log _{\epsilon }x}$ ${\displaystyle x(\log _{\epsilon }x-1)+C}$
${\displaystyle 0\cdot 4343\times x^{-1}}$ ${\displaystyle \log _{10}x}$ ${\displaystyle 0\cdot 4343x(\log _{\epsilon }x-1)+C}$
${\displaystyle a^{x}\log _{\epsilon }a}$ ${\displaystyle a^{x}}$ ${\displaystyle {\frac {a^{x}}{\log _{\epsilon }a}}+C}$
Trigonometrical.
${\displaystyle \cos x}$ ${\displaystyle \sin x}$ ${\displaystyle -\cos x+C}$
${\displaystyle -\sin x}$ ${\displaystyle \cos x}$ ${\displaystyle \sin x+C}$
${\displaystyle \sec ^{2}x}$ ${\displaystyle \tan x}$ ${\displaystyle -\log _{\epsilon }\cos x+C}$
Circular (Inverse).
${\displaystyle {\frac {1}{\sqrt {(1-x^{2})}}}}$ ${\displaystyle \arcsin x}$ ${\displaystyle x\cdot \arcsin x+{\sqrt {1-x^{2}}}+C}$
${\displaystyle -{\frac {1}{\sqrt {(1-x^{2})}}}}$ ${\displaystyle \arccos x}$ ${\displaystyle x\cdot \arccos x-{\sqrt {1-x^{2}}}+C}$
${\displaystyle {\frac {1}{1+x^{2}}}}$ ${\displaystyle \arctan x}$ ${\displaystyle x\cdot \arctan x-{\tfrac {1}{2}}\log _{\epsilon }(1+x^{2})+C}$
Hyperbolic.
${\displaystyle \cosh x}$ ${\displaystyle \sinh x}$ ${\displaystyle \cosh x+C}$
${\displaystyle \sinh x}$ ${\displaystyle \cosh x}$ ${\displaystyle \sinh x+C}$
${\displaystyle \operatorname {sech} ^{2}x}$ ${\displaystyle \tanh x}$ ${\displaystyle \log _{\epsilon }\cosh x+C}$
Miscellaneous.
${\displaystyle -{\frac {1}{(x+a)^{2}}}}$ ${\displaystyle {\frac {1}{x+a}}}$ ${\displaystyle \log _{\epsilon }(x+a)+C}$
${\displaystyle -{\frac {x}{(a^{2}+x^{2})^{\tfrac {3}{2}}}}}$ ${\displaystyle {\frac {1}{\sqrt {a^{2}+x^{2}}}}}$ ${\displaystyle \log _{\epsilon }(x+{\sqrt {a^{2}+x^{2}}})+C}$
${\displaystyle \mp {\frac {b}{(a\pm bx)^{2}}}}$ ${\displaystyle {\frac {1}{a\pm bx}}}$ ${\displaystyle \pm {\frac {1}{b}}\log _{\epsilon }(a\pm bx)+C}$
${\displaystyle {\frac {-3a^{x}x}{(a^{2}+x^{2})^{\tfrac {5}{2}}}}}$ ${\displaystyle {\frac {a^{2}}{(a^{2}+x^{2})^{\tfrac {3}{2}}}}}$ ${\displaystyle {\frac {x}{\sqrt {a^{2}+x^{2}}}}+C}$
${\displaystyle a\cdot \cos ax}$ ${\displaystyle \sin ax}$ ${\displaystyle {\frac {1}{a}}\cos ax+C}$
${\displaystyle -a\cdot \sin ax}$ ${\displaystyle \cos ax}$ ${\displaystyle {\frac {1}{a}}\sin ax+C}$
${\displaystyle a\cdot \sec ^{2}ax}$ ${\displaystyle \tan ax}$ ${\displaystyle -{\frac {1}{a}}\log _{\epsilon }\cos ax+C}$
${\displaystyle \sin 2x}$ ${\displaystyle \sin ^{2}x}$ ${\displaystyle {\frac {x}{2}}-{\frac {\sin 2x}{4}}+C}$
${\displaystyle -\sin 2x}$ ${\displaystyle \cos ^{2}x}$ ${\displaystyle {\frac {x}{2}}+{\frac {\sin 2x}{4}}+C}$
${\displaystyle n\cdot \sin ^{n-1}x\cdot \cos x}$ ${\displaystyle \sin ^{n}x}$ ${\displaystyle {\frac {\cos x}{n}}\sin ^{n-1}x+{\frac {n-1}{n}}\int \sin ^{n-2}xdx+C}$
${\displaystyle -{\frac {\cos x}{\sin ^{2}x}}}$ ${\displaystyle {\frac {1}{\sin x}}}$ ${\displaystyle \log _{\epsilon }\tan {\frac {x}{2}}+C}$
${\displaystyle -{\frac {\sin 2x}{\sin ^{4}x}}}$ ${\displaystyle {\frac {1}{\sin ^{2}x}}}$ ${\displaystyle -\operatorname {cotan} x+C}$
${\displaystyle {\frac {\sin ^{2}x-\cos ^{2}x}{\sin ^{2}x\cdot \cos ^{2}x}}}$ ${\displaystyle {\frac {1}{\sin x\cdot \cos x}}}$ ${\displaystyle \log _{\epsilon }\tan x+C}$
${\displaystyle n\cdot \sin mx\cdot \cos nx+m\cdot \sin nx\cdot \cos mx}$ ${\displaystyle \sin mx\cdot \sin nx}$ ${\displaystyle {\tfrac {1}{2}}\cos(m-n)x-{\tfrac {1}{2}}\cos(m+n)x+C}$
${\displaystyle 2a\cdot \sin 2ax}$ ${\displaystyle \sin ^{2}ax}$ ${\displaystyle {\frac {x}{2}}-{\frac {\sin 2ax}{4a}}+C}$
${\displaystyle -2a\cdot \sin 2ax}$ ${\displaystyle \cos ^{2}ax}$ ${\displaystyle {\frac {x}{2}}+{\frac {\sin 2ax}{4a}}+C}$