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[parent] general formulas for integration (Topic)
  1. $ \displaystyle \int f'(x)\,dx = f(x)+C$
  2. $ \displaystyle \int\lambda\,dx = \lambda x+C$
  3. $ \displaystyle \int\lambda f(x)\,dx = \lambda\int f(x)\,dx$
  4. $ \displaystyle \int(f(x)+g(x))\,dx = \int f(x)\,dx+\int g(x)\,dx$
  5. $ \displaystyle \int f(x)g'(x)\,dx = f(x)g(x)-\int g(x)f'(x)\,dx$
  6. $ \displaystyle \int g(f(x))f'(x)\,dx = G(f(x))+C$ if $ G'(t) = g(t)$
  7. $ \displaystyle \int [f(x)]^rf'(x)\,dx = \frac{1}{r\!+\!1}[f(x)]^{r+1}+C$ for $ r \neq -1$
  8. $ \displaystyle \int \frac{f'(x)}{f(x)}\,dx = \ln\vert f(x)\vert+C$
  9. $ \displaystyle \int e^{f(x)}f'(x)\,dx = e^{f(x)}+C$
  10. $ \displaystyle \int \!\frac{f(x)}{(f(x)\!+\!a)(f(x)\!+\!b)}\,dx \,=\, \frac{a}{a\!-\!b}\int\!\frac{dx}{f(x)\!+\!a}-\frac{b}{a\!-\!b}\int\!\frac{dx}{f(x)\!+\!b}$
  11. $ \displaystyle \int \sin(\omega x+\varphi)\,dx = -\frac{\cos(\omega x+\varphi)}{\omega}+C$
  12. $ \displaystyle \int \cos(\omega x+\varphi)\,dx = \frac{\sin(\omega x+\varphi)}{\omega}+C$
  13. $ \displaystyle \int \sinh(\omega x+\varphi)\,dx = \frac{\cosh(\omega x+\varphi)}{\omega}+C$
  14. $ \displaystyle \int \cosh(\omega x+\varphi)\,dx = \frac{\sinh(\omega x+\varphi)}{\omega}+C$
  15. $ \displaystyle \int \sqrt{ax\!+\!b}\;dx = \frac{2}{3a}(ax\!+\!b)\sqrt{ax\!+\!b}+C$
  16. $ \displaystyle \int \sqrt{ax^2\!+\!b}\;dx = \frac{x}{2}\sqrt{ax^2\!+\!b}+\frac{b}{2\sqrt{a}}\ln(x\sqrt{a}+\sqrt{ax^2\!+\!b})+C$

Some series-formed antiderivatives:

$ \displaystyle \int f(x)\,dx = C+f(0)x+\frac{f'(0)}{2!}x^2+\frac{f''(0)}{3!}x^3+\ldots$
$ \displaystyle \int f(x)\,dx = C+xf(x)-\frac{x^2}{2!}f'(x)+\frac{x^3}{3!}f''(x)-+\ldots$
$ \displaystyle \int UV\,dx = UV^{(-1)}-U'V^{(-2)}+U''V^{(-3)}-+\ldots \;=\; \sum_{n=0}^\infty (-1)^n U^{(n)}V^{(-n-1)}$

The derivatives with negative order mean that $ V$ has been integrated repeatedly.



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"general formulas for integration" is owned by pahio. [ full author list (3) ]
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See Also: table of derivatives, table of integrals, integration by parts

Other names:  integration formulas
Keywords:  antiderivative

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Cross-references: negative, derivatives, antiderivatives
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This is version 10 of general formulas for integration, born on 2007-12-04, modified 2007-12-06.
Object id is 10092, canonical name is GeneralFormulasForIntegration.
Accessed 885 times total.

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AMS MSC: 26A36 (Real functions :: Functions of one variable :: Antidifferentiation)
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