តារាងអាំងតេក្រាលនៃអនុគមន៍សនិទាន

ខាងក្រោមនេះជាតារាងអាំងតេក្រាល (ព្រីមីទីវ) នៃអនុគមន៍សនិទាន។ សូមមើល តារាងអាំងតេក្រាល សំរាប់បញ្ជីពេញលេញនៃគ្រប់អាំងតេក្រាល។

$\int (a x + b)^{n} d x$	$= \frac{(a x + b)^{n + 1}}{a (n + 1)}$ ទំព័រគំរូ:Spaces(ចំពោះ $n \neq - 1$ )
$\int \frac{c}{a x + b} d x$	$= \frac{c}{a} \ln \| a x + b \|$
$\int x (a x + b)^{n} d x$	$= \frac{a (n + 1) x - b}{a^{2} (n + 1) (n + 2)} (a x + b)^{n + 1}$ ទំព័រគំរូ:Spaces(ចំពោះ $n \in̸ {- 1, - 2}$ )

$\int \frac{x}{a x + b} d x$	$= \frac{x}{a} - \frac{b}{a^{2}} \ln \| a x + b \|$
$\int \frac{x}{(a x + b)^{2}} d x$	$= \frac{b}{a^{2} (a x + b)} + \frac{1}{a^{2}} \ln \| a x + b \|$
$\int \frac{x}{(a x + b)^{n}} d x$	$= \frac{a (1 - n) x - b}{a^{2} (n - 1) (n - 2) (a x + b)^{n - 1}}$ ទំព័រគំរូ:Spaces(ចំពោះ $n \in̸ {1, 2}$ )

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

\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 = {\begin{matrix} - \frac{1}{a} a r c t a n h \frac{x}{a} = \frac{1}{2 a} \ln \frac{a - x}{a + x} & (| 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} & (| x | > | a |) \end{matrix}

ចំពោះ $a \neq 0 :$

\int \frac{1}{a x^{2} + b x + c} d x = {\begin{matrix} \frac{2}{\sqrt{4 a c - b^{2}}} \arctan \frac{2 a x + b}{\sqrt{4 a c - b^{2}}} & (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}} | & (4 a c - b^{2} < 0) \\ - \frac{2}{2 a x + b} (4 a c - b^{2} = 0) \end{matrix}

\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 = {\begin{matrix} \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}}} & (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}} & (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)} & (4 a c - b^{2} = 0) \end{matrix}

\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 |]}

គ្រប់អនុគមន៍សនិទានទាំងអស់អាចធ្វើអាំងតេក្រាលបានដោយប្រើសមីការនិងអាំងតេក្រាលដោយផ្នែក ដោយបំបែកអនុគមន៍សនិទានជាផលបូកនៃអនុគមន៍ដែលមានទំរង់៖

\frac{e x + f}{{(a x^{2} + b x + c)}^{n}}

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