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وبلاگ زندگی با شیمی - انتگرال توابع اصم یا گنگ
تاریخ : شنبه 8 مرداد 1390 | 05:57 ب.ظ | نویسنده : محسن کمالی فر

\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


انتگرال 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


انتگرال توابع نمایی

\int e^x\,dx = e^x + C

\int a^x\,dx = \frac{a^x}{\ln{a}} + C


انتگرال 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

\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


انتگرال توابع هذلولوی

\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


انتگرال معکوس توابع هذلولوی

\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