جدول فرم‌های استاندارد

$\frac{d y}{d x}$	$⟵ y ⟶$	$\int y d x$
$1$	$x$	$\frac{1}{2} x^{2} + C$
$0$	$a$	$a x + C$
$1$	$x \pm a$	$\frac{1}{2} x^{2} \pm a x + C$
$a$	$a x$	$\frac{1}{2} a x^{2} + C$
$2 x$	$x^{2}$	$\frac{1}{3} x^{3} + C$
$n x^{n - 1}$	$x^{n}$	$\frac{1}{n + 1} x^{n + 1} + C$
$- x^{- 2}$	$x^{- 1}$	$\ln \| x \| + C$
$\frac{d u}{d x} \pm \frac{d v}{d x} \pm \frac{d w}{d x}$	$u \pm v \pm w$	$\int u d x \pm \int v d x \pm \int w d x$
$u \frac{d v}{d x} + v \frac{d u}{d x}$	$u v$	شکل کلی شناخته‌شده‌ای وجود ندارد
$\frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v^{2}}$	$\frac{u}{v}$	شکل کلی شناخته‌شده‌ای وجود ندارد
$\frac{d u}{d x}$	$u$	$u x - \int x d u + C$

نمایی و لگاریتمی:

$\frac{d y}{d x}$	$⟵ y ⟶$	$\int y d x$
$e^{x}$	$e^{x}$	$e^{x} + C$
$x^{- 1}$	$\ln x$	$x (\ln x - 1) + C$
$\frac{1}{x \ln b}$	$\log_{b} x$	$\frac{1}{\ln b} x (\ln x - 1) + C$
$a^{x} \ln a$	$a^{x}$	$\frac{a^{x}}{\ln a} + C$

مثلثاتی:

$\frac{d y}{d x}$	$⟵ y ⟶$	$\int y d x$
$\cos x$	$\sin x$	$- \cos x + C$
$- \sin x$	$\cos x$	$\sin x + C$
$\sec^{2} x$	$\tan x$	$- \ln \| \cos x \| + C$
$\sec x \tan x$	$\sec x$	$\ln \| \sec x + \tan x \| + C$

مثلثاتی معکوس:

$\frac{d y}{d x}$	$⟵ y ⟶$	$\int y d x$
$\frac{1}{\sqrt{1 - x^{2}}}$	$\arcsin x = \sin^{- 1} x$	$x \cdot \arcsin x + \sqrt{1 - x^{2}} + C$
$- \frac{1}{\sqrt{1 - x^{2}}}$	$\arccos x = \cos^{- 1} x$	$x \cdot \arccos x - \sqrt{1 - x^{2}} + C$
$\frac{1}{1 + x^{2}}$	$\arctan x = \tan^{- 1} x$	$x \cdot \arctan x - \frac{1}{2} \ln (1 + x^{2}) + C$

هذلولوی:

$\frac{d y}{d x}$	$⟵ y ⟶$	$\int y d x$
$\cosh x$	$\sinh x$	$\cosh x + C$
$\sinh x$	$\cosh x$	$\sinh x + C$
${sech}^{2} x$	$\tanh x$	$\ln \cosh x + C$
$- sech x \tanh x$	$sech x$	$2 \arctan (e^{x}) + C$

هذلولوی معکوس:

$\frac{d y}{d x}$	$⟵ y ⟶$	$\int y d x$
$\frac{1}{\sqrt{x^{2} + 1}}$	$arcsinh x = \sinh^{- 1} x$	$- \sqrt{1 + x^{2}} + x arcsinh x + C$
$\frac{1}{\sqrt{x^{2} - 1}}$	$arccosh x = \cosh^{- 1} x$	$- \sqrt{x^{2} - 1} + x arccosh x + C$
$\frac{1}{1 - x^{2}}$	$arctanh x = \tanh^{- 1} x$	$x arctanh x + \frac{1}{2} \ln \| 1 - x^{2} \| + C$

متفرقه:

$\frac{d y}{d x}$	$⟵ y ⟶$	$\int y d x$
$- \frac{1}{(x + a)^{2}}$	$\frac{1}{x + a}$	$\ln \| x + a \| + C$
$\frac{x}{\sqrt{x^{2} + a^{2}}}$	$\sqrt{a^{2} + x^{2}}$	$\frac{1}{2} x \sqrt{x^{2} + a^{2}} + \frac{a^{2}}{2} \ln (x + \sqrt{x^{2} + a^{2}}) + C$
$- \frac{x}{(a^{2} + x^{2})^{\frac{3}{2}}}$	$\frac{1}{\sqrt{a^{2} + x^{2}}}$	$\ln (x + \sqrt{a^{2} + x^{2}}) + C$
$\mp \frac{b}{(a \pm b x)^{2}}$	$\frac{1}{a \pm b x}$	$\pm \frac{1}{b} \ln \| a \pm b x \| + C$
$- \frac{3 a^{2} x}{(a^{2} + x^{2})^{\frac{5}{2}}}$	$\frac{a^{2}}{(a^{2} + x^{2})^{\frac{3}{2}}}$	$\frac{x}{\sqrt{a^{2} + x^{2}}} + C$
$a \cdot \cos a x$	$\sin a x$	$- \frac{1}{a} \cos a x + C$
$- a \cdot \sin a x$	$\cos a x$	$\frac{1}{a} \sin a x + C$
$a \cdot \sec^{2} a x$	$\tan a x$	$- \frac{1}{a} \ln \| \cos a x \| + C$
$\sin 2 x$	$\sin^{2} x$	$\frac{x}{2} - \frac{\sin 2 x}{4} + C$
$- \sin 2 x$	$\cos^{2} x$	$\frac{x}{2} + \frac{\sin 2 x}{4} + C$
$n \cdot \sin^{n - 1} x \cdot \cos x$	$\sin^{n} x$	$- \frac{\cos x}{n} \sin^{n - 1} x + \frac{n - 1}{n} \int \sin^{n - 2} x d x$
$- \frac{\cos x}{\sin^{2} x}$	$\frac{1}{\sin x}$	$\ln \| \tan \frac{x}{2} \| + C$
$- \frac{\sin 2 x}{\sin^{4} x}$	$\frac{1}{\sin^{2} x}$	$- \cot x + C$
$\frac{\sin^{2} x - \cos^{2} x}{\sin^{2} x \cdot \cos^{2} x}$	$\frac{1}{\sin x \cdot \cos x}$	$\ln \| \tan x \| + C$
$n \cdot \sin m x \cdot \cos n x + m \cdot \sin n x \cdot \cos m x$	$\sin m x \cdot \sin n x$	$\frac{1}{2} \cos (m - n) x - \frac{1}{2} \cos (m + n) x + C$
$2 a \cdot \sin 2 a x$	$\sin^{2} a x$	$\frac{x}{2} - \frac{\sin 2 a x}{4 a} + C$
$- 2 a \cdot \sin 2 a x$	$\cos^{2} a x$	$\frac{x}{2} + \frac{\sin 2 a x}{4 a} + C$

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