ملحق:قائمة تكاملات الدوال النسبية

$\int \frac{x^{2}}{a x + b} d x$	$= \frac{1}{a^{3}} (\frac{(a x + b)^{2}}{2} - 2 b (a x + b) + b^{2} \ln \| a x + b \|)$
$\int \frac{x^{2}}{(a x + b)^{2}} d x$	$= \frac{1}{a^{3}} (a x + b - 2 b \ln \| a x + b \| - \frac{b^{2}}{a x + b})$
$\int \frac{x^{2}}{(a x + b)^{3}} d x$	$= \frac{1}{a^{3}} (\ln \| a x + b \| + \frac{2 b}{a x + b} - \frac{b^{2}}{2 (a x + b)^{2}})$
$\int \frac{x^{2}}{(a x + b)^{n}} d x$	$= \frac{1}{a^{3}} (- \frac{(a x + b)^{3 - n}}{(n - 3)} + \frac{2 b (a + b)^{2 - n}}{(n - 2)} - \frac{b^{2} (a x + b)^{1 - n}}{(n - 1)}) (for n \in̸ {1, 2, 3})$

$\int \frac{1}{x (a x + b)} d x$	$= - \frac{1}{b} \ln \| \frac{a x + b}{x} \|$
$\int \frac{1}{x^{2} (a x + b)} d x$	$= - \frac{1}{b x} + \frac{a}{b^{2}} \ln \| \frac{a x + b}{x} \|$
$\int \frac{1}{x^{2} (a x + b)^{2}} d x$	$= - a (\frac{1}{b^{2} (a x + b)} + \frac{1}{a b^{2} x} - \frac{2}{b^{3}} \ln \| \frac{a x + b}{x} \|)$
$\int \frac{1}{x^{2} + a^{2}} d x$	$= \frac{1}{a} \arctan \frac{x}{a}$
$\int \frac{1}{x^{2} - a^{2}} d x =$	$- \frac{1}{a} a r c t a n h \frac{x}{a} = \frac{1}{2 a} \ln \frac{a - x}{a + x} (for \| x \| < \| a \|)$
	$- \frac{1}{a} a r c c o t h \frac{x}{a} = \frac{1}{2 a} \ln \frac{x - a}{x + a} (for \| x \| > \| a \|)$

for $a \neq 0 :$

$\int \frac{1}{a x^{2} + b x + c} d x =$	$\frac{2}{\sqrt{4 a c - b^{2}}} \arctan \frac{2 a x + b}{\sqrt{4 a c - b^{2}}} (for 4 a c - b^{2} > 0)$
	$- \frac{2}{\sqrt{b^{2} - 4 a c}} a r c t a n h \frac{2 a x + b}{\sqrt{b^{2} - 4 a c}} = \frac{1}{\sqrt{b^{2} - 4 a c}} \ln \| \frac{2 a x + b - \sqrt{b^{2} - 4 a c}}{2 a x + b + \sqrt{b^{2} - 4 a c}} \| (for 4 a c - b^{2} < 0)$
	$- \frac{2}{2 a x + b} (for 4 a c - b^{2} = 0)$
$\int \frac{x}{a x^{2} + b x + c} d x$	$= \frac{1}{2 a} \ln \| a x^{2} + b x + c \| - \frac{b}{2 a} \int \frac{d x}{a x^{2} + b x + c}$

$\int \frac{m x + n}{a x^{2} + b x + c} d x =$	$\frac{m}{2 a} \ln \| a x^{2} + b x + c \| + \frac{2 a n - b m}{a \sqrt{4 a c - b^{2}}} \arctan \frac{2 a x + b}{\sqrt{4 a c - b^{2}}} (for 4 a c - b^{2} > 0)$
	$\frac{m}{2 a} \ln \| a x^{2} + b x + c \| - \frac{2 a n - b m}{a \sqrt{b^{2} - 4 a c}} a r c t a n h \frac{2 a x + b}{\sqrt{b^{2} - 4 a c}} (for 4 a c - b^{2} < 0)$
	$\frac{m}{2 a} \ln \| a x^{2} + b x + c \| - \frac{2 a n - b m}{a (2 a x + b)} (for 4 a c - b^{2} = 0)$

\int \frac{1}{(a x^{2} + b x + c)^{n}} d x = \frac{2 a x + b}{(n - 1) (4 a c - b^{2}) (a x^{2} + b x + c)^{n - 1}} + \frac{(2 n - 3) 2 a}{(n - 1) (4 a c - b^{2})} \int \frac{1}{(a x^{2} + b x + c)^{n - 1}} d x

\int \frac{x}{(a x^{2} + b x + c)^{n}} d x = \frac{b x + 2 c}{(n - 1) (4 a c - b^{2}) (a x^{2} + b x + c)^{n - 1}} - \frac{b (2 n - 3)}{(n - 1) (4 a c - b^{2})} \int \frac{1}{(a x^{2} + b x + c)^{n - 1}} d x

\int \frac{1}{x (a x^{2} + b x + c)} d x = \frac{1}{2 c} \ln | \frac{x^{2}}{a x^{2} + b x + c} | - \frac{b}{2 c} \int \frac{1}{a x^{2} + b x + c} d x

\int \frac{d x}{x^{2^{n}} + 1} = \sum_{k = 1}^{2^{n - 1}} {\frac{1}{2^{n - 1}} [\sin (\frac{(2 k - 1) π}{2^{n}}) \arctan [(x - \cos (\frac{(2 k - 1) π}{2^{n}})) \csc (\frac{(2 k - 1) π}{2^{n}})]] - \frac{1}{2^{n}} [\cos (\frac{(2 k - 1) π}{2^{n}}) \ln | x^{2} - 2 x \cos (\frac{(2 k - 1) π}{2^{n}}) + 1 |]}

بوابة رياضيات تصفح مقالات ويكيبيديا المهتمة بالرياضيات.

$\int \frac{m x + n}{a x^{2} + b x + c} d x =$	$\frac{m}{2 a} \ln \| a x^{2} + b x + c \| + \frac{2 a n - b m}{a \sqrt{4 a c - b^{2}}} \arctan \frac{2 a x + b}{\sqrt{4 a c - b^{2}}} (for 4 a c - b^{2} > 0)$
	$\frac{m}{2 a} \ln \| a x^{2} + b x + c \| - \frac{2 a n - b m}{a \sqrt{b^{2} - 4 a c}} a r c t a n h \frac{2 a x + b}{\sqrt{b^{2} - 4 a c}} (for 4 a c - b^{2} < 0)$
	$\frac{m}{2 a} \ln \| a x^{2} + b x + c \| - \frac{2 a n - b m}{a (2 a x + b)} (for 4 a c - b^{2} = 0)$

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