جدول التكاملات الأساسية

$\int {x^n } \,dx = \frac{{x^{n + 1} }}{{n + 1}} + C\quad n \ne -1$

$\int {x^{ - 1} } \,dx = \ln x + C$

دوال مثلثية

$\int {\sin x} \,dx = - \cos x + C$

$\int {\csc } x\,dx = \ln \left| {\csc x - \cot x} \right| + C$

$\int {\cos x} \,dx = \sin x + C$

$\int {\sec } x\,\tan x\,dx = \sec x + C$

$\int {\tan x} \,dx = - \ln \left| {\cos x} \right| + C = \ln \left| {\sec x} \right| + C$

$\int {\csc } x\,\cot x\,dx = - \csc x + C$

$\int {\cot x} \,dx = \ln \left| {\sin x} \right| + C = - \ln \left| {\csc x} \right| + C$

$\int {\sec ^2 } x\,dx = \tan x + C$

$\int {\sec } x\,dx = \ln \left| {\sec x + \tan x} \right| + C$

$\int {\csc ^2 } x\,dx = - \cot x + C$

دوال مثلثية عكسية

$\int {\arcsin } x\,dx = x\arcsin x + \sqrt {1 - x^2 } + C$

$\int {{\mathop{\rm arccsc}\nolimits} } x\,dx = x{\mathop{\rm arccsc}\nolimits} x + \ln \left| {x + x\sqrt {\frac{{x^2 - 1}}{{x^2 }}} } \right| + C$

$\int {\arccos } x\,dx = x\arccos x - \sqrt {1 - x^2 } + C$

$\int {{\mathop{\rm arcsec}\nolimits} } x\,dx = x{\mathop{\rm arcsec}\nolimits} x - \ln \left| {x + x\sqrt {\frac{{x^2 - 1}}{{x^2 }}} } \right| + C$

$\int {\arctan } x\,dx = x\arctan x - \frac{1}{2}\ln |1 + x^2 | + C$

$\int {{\mathop{\rm arccot}\nolimits} } x\,dx = x{\mathop{\rm arccot}\nolimits} x + \frac{1}{2}\ln |1 + x^2 | + C$

الدوال الزائدية

$\int {\sinh } x\,dx = \cosh x + C$

$\int {{\rm{csch}}} \,x\,dx = \ln \left| {\tanh \frac{x}{2}} \right| + C$

$\int {\cosh } x\,dx = \sinh x + C$

$\int {\coth } x\,dx = \ln |\sinh x| + C$

$\int {\tanh } x\,dx = \ln |\cosh x| + C$

$\int {{\rm{sech}}^2 } x\,dx = \tanh x + C$

$\int {{\rm{sech}}} \,x\,dx = \arctan (\sinh x) + C$

$\int {{\rm{csch}}^2 } x\,dx = - \coth x + C$

الدوال الزايدية العكسية

$\int {{\mathop{\rm arcsinh}\nolimits} } x\,dx = x{\mathop{\rm arcsinh}\nolimits} x - \sqrt {x^2 + 1} + C$

$\int {{\mathop{\rm arccosh}\nolimits} } x\,dx = x{\mathop{\rm arccosh}\nolimits} x - \sqrt {x^2 - 1} + C$

$\int {{\mathop{\rm arctanh}\nolimits} } x\,dx = x{\mathop{\rm arctanh}\nolimits} x + \frac{1}{2}\log (1 - x^2 ) + C$

$\int {{\mathop{\rm arccsch}\nolimits} } \,x\,dx = x{\mathop{\rm arccsch}\nolimits} x + \log \left[ {x\left( {\sqrt {1 + \frac{1}{{x^2 }}} + 1} \right)} \right] + C$

$\int {{\mathop{\rm arcsech}\nolimits} } \,x\,dx = x{\mathop{\rm arcsech}\nolimits} x - \arctan \left( {\frac{x}{{x - 1}}\sqrt {\frac{{1 - x}}{{1 + x}}} } \right) + C$

$\int {{\mathop{\rm arccoth}\nolimits} } \,x\,dx = x{\mathop{\rm arccoth}\nolimits} x + \frac{1}{2}\log (x^2 - 1) + C$