1.24: Table of Standard Forms
- Page ID
- 146260
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Table of Standard Forms
| \(\dfrac{dy}{dx}\) | \(y\) | \(\int y\, dx\) |
|---|---|---|
| Algebraic. | ||
| \(1\) | \(x\) | \(\frac{1}{2} x^2 + C\) |
| \(0\) | \(a\) | \(ax + C \) |
| \(1\) | \(x ± a\) | \(\frac{1}{2} x^2 ± ax + C\) |
| \(a\) | \(ax \) | \(\frac{1}{2} ax^2 + C \) |
| \(2x\) | \(x^2\) | \(\frac{1}{3} x^3 + C \) |
| \(nx^{n-1}\) | \(x^n\) | \( \dfrac{1}{n+1} x^{n+1} + C \) |
| \(-x^{-2} \) | \(x^{-1}\) | \(\log_\epsilon x + C\) |
| \(\dfrac{du}{dx} ± \dfrac{dv}{dx} ± \dfrac{dw}{dx}\) | \(u ± v ± w\) | \(\int u\, dx ± \int v\, dx ± \int w\, dx\) |
| \(u\, \dfrac{dv}{dx} + v\, \dfrac{du}{dx}\) | \(uv\) | No general form known |
| \(\dfrac{v\, \dfrac{du}{dx} - u\, \dfrac{dv}{dx}}{v^2}\) | \(\dfrac{u}{v}\) | No general form known |
| \(\dfrac{du}{dx}\) | \(u\) | \(ux - \int x\, du + C\) |
| Exponential and Logarithmic. | ||
| \(\epsilon^x\) | \(\epsilon^x\) | \(\epsilon^x + C\) |
| \(x^{-1}\) | \(\log_\epsilon x\) | \( x(\log_\epsilon x - 1) + C\) |
| \(0.4343 × x^{-1}\) | \(\log_{10} x\) | \(0.4343x (\log_\epsilon x - 1) + C\) |
| \(a^x \log_\epsilon a\) | \(a^x\) | \(\dfrac{a^x}{\log_\epsilon a} + C\) |
| 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). | ||
| \(\dfrac{1}{\sqrt{(1-x^2)}}\) | \(\arcsin x\) | \(x · \arcsin x + \sqrt{1 - x^2} + C\) |
| \(-\dfrac{1}{\sqrt{(1-x^2)}}\) | \(\arccos x\) | \(x · \arccos x - \sqrt{1 - x^2} + C\) |
| \(\dfrac{1}{1+x^2}\) | \(\arctan x\) | \(x · \arctan x - \frac{1}{2} \log_\epsilon (1 + x^2) + C\) |
| Hyperbolic. | ||
| \(\cosh x \) | \(\sinh x\) | \(\cosh x + C\) |
| \(\sinh x \) | \(\cosh x\) | \(\sinh x + C\) |
| \(\text{sech}^2 x \) | \(\tanh x\) | \(\log_\epsilon \cosh x + C \) |
| Miscellaneous. | ||
| \(-\dfrac{1}{(x + a)^2}\) | \(\dfrac{1}{x + a}\) | \( \log_\epsilon (x+a) + C \) |
| \(-\dfrac{x}{(a^2 + x^2)^{\frac{3}{2}}}\) | \(\dfrac{1}{\sqrt{a^2 + x^2}}\) | \(\log_\epsilon (x + \sqrt{a^2 + x^2}) + C \) |
| \(\mp \dfrac{b}{(a ± bx)^2}\) | \(\dfrac{1}{a ± bx}\) | \(± \dfrac{1}{b} \log_\epsilon (a ± bx) + C \) |
| \(-\dfrac{3a^2x}{(a^2 + x^2)^{\frac{5}{2}}}\) | \(\dfrac{a^2}{(a^2 + x^2)^{\frac{3}{2}}}\) | \(\dfrac{x}{\sqrt{a^2 + x^2}} + C \) |
| \( a · \cos ax\) | \(\sin ax\) | \(-\dfrac{1}{a} \cos ax + C \) |
| \(-a · \sin ax\) | \(\cos ax\) | \( \dfrac{1}{a} \sin ax + C \) |
| \( a · \sec^2ax\) | \(\tan ax\) | \(-\dfrac{1}{a} \log_\epsilon \cos ax + C \) |
| \( \sin 2x\) | \(\sin^2 x\) | \(\dfrac{x}{2} - \dfrac{\sin 2x}{4} + C \) |
| \(-\sin 2x\) | \(\cos^2 x\) | \(\dfrac{x}{2} + \dfrac{\sin 2x}{4} + C \) |
| \(n · \sin^{n-1} x · \cos x\) | \( \sin^n x\) | \(-\dfrac{\cos x}{n} \sin ^{n-1} x+\dfrac{n-1}{n} \int \sin ^{n-2} x d x+C\) |
| \(-\dfrac{\cos x}{\sin^2 x}\) | \(\dfrac{1}{\sin x}\) | \(\log_\epsilon \tan \dfrac{x}{2} + C\) |
| \(-\dfrac{\sin 2x}{\sin^4 x}\) | \(\dfrac{1}{\sin^2 x}\) | \( -\text{cotan} x + C\) |
| \(\dfrac{\sin^2 x - \cos^2 x}{\sin^2 x · \cos^2 x}\) | \( \dfrac{1}{\sin x · \cos x}\) | \( \log_\epsilon \tan x + C \) |
| \(n · \sin mx · \cos nx + m · \sin nx · \cos mx \) | \(\sin mx · \sin nx\) | \(\frac{1}{2} \cos(m - n)x - \frac{1}{2} \cos(m + n)x + C\) |
| \( 2a·\sin 2ax\) | \(\sin^2 ax\) | \(\dfrac{x}{2} - \dfrac{\sin 2ax}{4a} + C \) |
| \(-2a·\sin 2ax\) | \(\cos^2 ax\) | \(\dfrac{x}{2} + \dfrac{\sin 2ax}{4a} + C \) |