# Calculus Made Easy/Table of Standard Forms

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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