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\int (a x + b)^{n} d x = {\begin{matrix} \frac{(a x + b)^{n + 1}}{a (n + 1)}, & n \neq - 1 \\ \frac{1}{a} \ln | a x + b |, & n = - 1 \end{matrix}

\int x (a x + b)^{n} d x = {\begin{matrix} \frac{a (n + 1) x - b}{a^{2} (n + 1) (n + 2)} (a x + b)^{n + 1}, & n \in̸ {- 1, - 2} \\ \frac{x}{a} - \frac{b}{a^{2}} \ln | a x + b |, & n = - 1 \\ \frac{b}{a^{2} (a x + b)} + \frac{1}{a^{2}} \ln | a x + b |, & n = - 2 \end{matrix}

\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}}, n \in̸ {1, 2}

\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}}) arctg [(x - \cos (\frac{(2 k - 1) π}{2^{n}})) cosec (\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 |]} + C

\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{1}{(n - 3) (a x + b)^{n - 3}} + \frac{2 b}{(n - 2) (a x + b)^{n - 2}} - \frac{b^{2}}{(n - 1) (a x + b)^{n - 1}}),

для

n \in̸ {1, 2, 3}

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

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

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

\int \frac{d x}{(x^{2} + a^{2})^{2}} = \frac{x}{2 a^{2} (x^{2} + a^{2})} + \frac{1}{2 a^{3}} arctg \frac{x}{a}

\int \frac{d x}{(x^{2} + a^{2})^{3}} = \frac{x}{4 a^{2} (x^{2} + a^{2})^{2}} + \frac{3 x}{8 a^{4} (x^{2} + a^{2})} + \frac{3}{8 a^{5}} arctg \frac{x}{a}

\int \frac{d x}{x^{2} - a^{2}} = - \frac{1}{a} arth \frac{x}{a} = \frac{1}{2 a} \ln \frac{a - x}{a + x},

для

| x | < | a |

\int \frac{d x}{x^{2} - a^{2}} = - \frac{1}{a} arcth \frac{x}{a} = \frac{1}{2 a} \ln \frac{x - a}{x + a},

для

| x | > | a |

\int \frac{d x}{a x^{2} + b x + c} = \frac{2}{\sqrt{4 a c - b^{2}}} arctg \frac{2 a x + b}{\sqrt{4 a c - b^{2}}},

для

4 a c - b^{2} > 0

\int \frac{d x}{a x^{2} + b x + c} = - \frac{2}{\sqrt{b^{2} - 4 a c}} arth \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}} |,

для

4 a c - b^{2} < 0

\int \frac{d x}{a x^{2} + b x + c} = - \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}}} arctg \frac{2 a x + b}{\sqrt{4 a c - b^{2}}},

для

4 a c - b^{2} > 0

\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{b^{2} - 4 a c}} arth \frac{2 a x + b}{\sqrt{b^{2} - 4 a c}},

для

4 a c - b^{2} < 0

\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 (2 a x + b)},

для

4 a c - b^{2} = 0

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

\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{d x}{(a x^{2} + b x + c)^{n - 1}}

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

Библиографи

Градштейн И. С. Рыжик И. М. Таблицы интегралов, сумм, рядов и произведений. — 4-е изд. — М.: Наука, 1963. — ISBN 0-12-294757-6 // EqWorld
Двайт Г. Б. Таблицы интегралов СПб: Издательство и типография АО ВНИИГ им. Б. В. Веденеева, 1995. — 176 с. — ISBN 5-85529-029-8.
D. Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st ed., 2002. ISBN 1-58488-291-3.
M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 1964. ISBN 0-486-61272-4