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جدول مختصر انتگرالها - ویکی‌پدیا

جدول مختصر انتگرالها

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جدول مختصر انتگرال ها

فهرست مندرجات

[ویرایش] مختصر

1.\int u.dv=uv- \int v.du

2.\int a^u.du=\frac{(a^u)}{Ln a}+c,a\ne1,a>1


3.\int cosu=sinu+c


4.\int sinu=cosu+c


5.\int (ax+b)^n.dx=\frac{(ax+b)^{n+1})}{a(n+1)}+c,n\ne-1


6.\int (ax+b)^{-1}.dx=\frac{1}{a}ln\mid ax+b\mid+c


7.\int x(ax+b)^n.dx=\frac{(ax+b)^{n+1})}{a^2} . [\frac{ax+b}{n+2} - \frac{b}{n+1} ]+c,n\ne -1,-2


8.\int x(ax+b)^{-1}.dx=\frac{x}{a}-\frac{b}{a^2} ln\mid ax+b\mid+c


9.\int x(ax+b)^{-2}=\frac{1}{a^2}[ln\mid ax+b\mid+\frac{b}{ax+b}]+c


10.\int \frac {dx}{x(ax+b)}=\frac {1}{b} ln \mid \frac{x}{ax+b}\mid +c


[جدول کامل انتگرال ها: http://en.wikipedia.org/wiki/Table_of_integrals]

[ویرایش] عمومی

\int af(x)\,dx = a\int f(x)\,dx
\int [f(x) \pm g(x)]\,dx = \int f(x)\,dx \pm \int g(x)\,dx
\int f'(x)g(x)\,dx = f(x)g(x) - \int f(x)g'(x)\,dx
\int  {f'(x)\over f(x)}\,dx= \ln{\left|f(x)\right|} + C
\int  {f'(x) f(x)}\,dx= {1 \over 2} [ f(x) ]^2 + C
\int [f(x)]^n f'(x)\,dx = {[f(x)] \over n+1}^{n+1} + C \qquad\mbox{(for } n\neq -1\mbox{)}\,\!
\int(k)dx=kx
\int_{a}^{b}(u)dx=\int_{a}(u)dx-\int_{b}(u)dx+C
\int(m+n-u)dx=\int(m)dx+\int(n)dx-\int(u)dx+C

[ویرایش] توابع گویا

\int \,dx = x + C
\int (x^n)\,dx =  \frac{x^{n+1}}{n+1} + C\qquad\mbox{ where }n \ne -1
\int (\frac{1}{x})\,dx = \int (x^{-1})\,dx = \ln{\left|x\right|} + C
\int (\frac{1}{a^2+x^2})dx = {1 \over a}\arctan {x \over a} + C
more integrals: List of integrals of irrational functions
\int {dx \over \sqrt{a^2-x^2}} = \sin^{-1} {x \over a} + C
\int {-dx \over \sqrt{a^2-x^2}} = \cos^{-1} {x \over a} + C
\int {dx \over x \sqrt{x^2-a^2}} = {1 \over a} \sec^{-1} {|x| \over a} + C
\int {-dx \over x \sqrt{x^2-a^2}} = {1 \over a} \csc^{-1} {|x| \over a} + C

[ویرایش] توابع لگاریتمی

\int \ln {x}\,dx = x \ln {x} - x + C
\int \log_b {x}\,dx = x\log_b {x} - x\log_b {e} + C

[ویرایش] توابع نمایی

more integrals: List of integrals of exponential functions
\int e^x\,dx = e^x + C
\int a^x\,dx = \frac{a^x}{\ln{a}} + C

[ویرایش] توابع مثلثاتی

\int \sin{x}\, dx = -\cos{x} + C
\int \cos{x}\, dx = \sin{x} + C
\int \tan{x} \, dx = -\ln{\left| \cos {x} \right|} + C
\int \cot{x} \, dx = \ln{\left| \sin{x} \right|} + C
\int \sec{x} \, dx = \ln{\left| \sec{x} + \tan{x}\right|} + C
\int \csc{x} \, dx = -\ln{\left| \csc{x} + \cot{x}\right|} + C
\int \sec^2 x \, dx = \tan x + C
\int \csc^2 x \, dx = -\cot x + C
\int \sec{x} \, \tan{x} \, dx = \sec{x} + C
\int \csc{x} \, \cot{x} \, dx = -\csc{x} + C
\int \sin^2 x \, dx = \frac{1}{2}(x - \sin x \cos x) + C
\int \cos^2 x \, dx = \frac{1}{2}(x + \sin x \cos x) + C
\int \sec^3 x \, dx = \frac{1}{2}\sec x \tan x + \frac{1}{2}\ln|\sec x + \tan x| + C
(see integral of secant cubed)
\int \sin^n x \, dx = - \frac{\sin^{n-1} {x} \cos {x}}{n} + \frac{n-1}{n} \int \sin^{n-2}{x} \, dx
\int \cos^n x \, dx = \frac{\cos^{n-1} {x} \sin {x}}{n} + \frac{n-1}{n} \int \cos^{n-2}{x} \, dx
\int \arctan{x} \, dx = x \, \arctan{x} - \frac{1}{2} \ln{\left| 1 + x^2\right|} + C

[ویرایش] توابع هایپربولیک

\int \sinh x \, dx = \cosh x + C
\int \cosh x \, dx = \sinh x + C
\int \tanh x \, dx = \ln| \cosh x | + C
\int \mbox{csch}\,x \, dx = \ln\left| \tanh {x \over2}\right| + C
\int \mbox{sech}\,x \, dx = \arctan(\sinh x) + C
\int \coth x \, dx = \ln| \sinh x | + C
\int \mbox{sech}^2 x\, dx = \tanh x + C

[ویرایش] Inverse hyperbolic functions

\int \operatorname{arcsinh}\, x \, dx  = x\, \operatorname{arcsinh}\, x - \sqrt{x^2+1} + C
\int \operatorname{arccosh}\, x \, dx  = x\, \operatorname{arccosh}\, x - \sqrt{x^2-1} + C
\int \operatorname{arctanh}\, x \, dx  = x\, \operatorname{arctanh}\, x + \frac{1}{2}\log{(1-x^2)} + C
\int \operatorname{arccsch}\,x \, dx = x\, \operatorname{arccsch}\, x+ \log{\left[x\left(\sqrt{1+\frac{1}{x^2}} + 1\right)\right]} + C
\int \operatorname{arcsech}\,x \, dx = x\, \operatorname{arcsech}\, x- \arctan{\left(\frac{x}{x-1}\sqrt{\frac{1-x}{1+x}}\right)} + C
\int \operatorname{arccoth}\,x \, dx  = x\, \operatorname{arccoth}\, x+ \frac{1}{2}\log{(x^2-1)} + C
\int_0^\infty{\sqrt{x}\,e^{-x}\,dx} = \frac{1}{2}\sqrt \pi (see also Gamma function)
\int_0^\infty{e^{-x^2}\,dx} = \frac{1}{2}\sqrt \pi (the Gaussian integral)
\int_0^\infty{\frac{x}{e^x-1}\,dx} = \frac{\pi^2}{6} (see also Bernoulli number)
\int_0^\infty{\frac{x^3}{e^x-1}\,dx} = \frac{\pi^4}{15}
\int_0^\infty\frac{\sin(x)}{x}\,dx=\frac{\pi}{2}
\int_0^\frac{\pi}{2}\sin^n{x}\,dx=\int_0^\frac{\pi}{2}\cos^n{x}\,dx=\frac{1 \cdot 3 \cdot 5 \cdot \cdots \cdot (n-1)}{2 \cdot 4 \cdot 6 \cdot \cdots \cdot n}\frac{\pi}{2} (if n is an even integer and   \scriptstyle{n \ge 2})
\int_0^\frac{\pi}{2}\sin^n{x}\,dx=\int_0^\frac{\pi}{2}\cos^n{x}\,dx=\frac{2 \cdot 4 \cdot 6 \cdot \cdots \cdot (n-1)}{3 \cdot 5 \cdot 7 \cdot \cdots \cdot n} (if  \scriptstyle{n} is an odd integer and   \scriptstyle{n \ge 3} )
\int_0^\infty\frac{\sin^2{x}}{x^2}\,dx=\frac{\pi}{2}
\int_0^\infty  x^{z-1}\,e^{-x}\,dx = \Gamma(z) (where Γ(z) is the Gamma function)
\int_{-\infty}^\infty e^{-(ax^2+bx+c)}\,dx=\sqrt{\frac{\pi}{a}}\exp\left[\frac{b^2-4ac}{4a}\right] (where exp[u] is the exponential function eu, and a > 0)
\int_{0}^{2 \pi} e^{x \cos \theta} d \theta = 2 \pi I_{0}(x) (where I0(x) is the modified Bessel function of the first kind)
\int_{0}^{2 \pi} e^{x \cos \theta + y \sin \theta} d \theta = 2 \pi I_{0} \left(\sqrt{x^2 + y^2}\right)
\int_{-\infty}^{\infty}{(1 + x^2/\nu)^{-(\nu + 1)/2}dx} = \frac { \sqrt{\nu \pi} \ \Gamma(\nu/2)} {\Gamma((\nu + 1)/2))}\,, \nu > 0\,, this is related to the probability density function of the Student's t-distribution)

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