Неопределенный интеграл

Множество первообразных функций f(x) называется неопределенным интегралом от этой функции и обозначается символом $begin mathsize 12px style integral f open parentheses x close parentheses d x end style$ .

Как следует из изложенного выше, если F(x) - некоторая первообразная функция f(x), то $begin mathsize 12px style integral f open parentheses x close parentheses d x equals F open parentheses x close parentheses plus C end style$ , где С - произвольная постоянная.

Функцию f(x) принято называть подынтегральной функцией, произведение f(x)dx - подынтегральным выражением.

Свойства неопределенного интеграла, непосредственно следующие из определения:

$begin mathsize 12px style d integral f open parentheses x close parentheses d x equals f open parentheses x close parentheses d x end style$
$begin mathsize 12px style integral F apostrophe open parentheses x close parentheses d x equals F open parentheses x close parentheses plus C end style$ $open parentheses size 12px и size 12px л size 12px и size 12px space size 12px integral size 12px d size 12px F open parentheses size 12px x close parentheses size 12px equals size 12px F open parentheses size 12px x close parentheses size 12px plus size 12px C close parentheses$

Таблица неопределенных интегралов.

$begin mathsize 12px style integral 0 asterisk times d x equals C end style$
$begin mathsize 12px style integral 1 asterisk times d x equals x plus C end style$
$begin mathsize 12px style integral x to the power of alpha d x equals fraction numerator x to the power of alpha plus 1 end exponent over denominator alpha plus 1 end fraction plus C space end style$ $open parentheses size 12px alpha size 12px not equal to size 12px minus size 12px 1 close parentheses$
$begin mathsize 12px style integral 1 over x d x equals integral fraction numerator d x over denominator x end fraction equals ln open vertical bar x close vertical bar plus C end style$
$begin mathsize 12px style integral a to the power of x d x equals fraction numerator a to the power of x over denominator ln a end fraction plus C semicolon end style$ $begin mathsize 12px style integral e to the power of x d x equals e to the power of x plus C end style$
$begin mathsize 12px style integral sin space x d x equals negative cos space x plus C end style$
$begin mathsize 12px style integral c o s space x d x equals sin space x plus C end style$
$begin mathsize 12px style integral fraction numerator d x over denominator sin squared x end fraction equals negative c t g space x plus С end style$
$begin mathsize 12px style integral fraction numerator d x over denominator c o s squared x end fraction equals negative t g space x plus С end style$
$begin mathsize 12px style integral s h space x d x equals c h space x plus C end style$
$begin mathsize 12px style integral c h space x d x equals s h space x plus C end style$
$begin mathsize 12px style integral fraction numerator d x over denominator s h squared x end fraction equals negative c t h space x plus C end style$
$begin mathsize 12px style integral fraction numerator d x over denominator c h squared x end fraction equals t h space x plus C end style$
$begin mathsize 12px style integral fraction numerator 1 over denominator x squared plus a squared end fraction d x equals end style$ $begin mathsize 12px style integral fraction numerator d x over denominator x squared plus a squared end fraction equals end style$ $begin mathsize 12px style 1 over a a r c t g x over a plus C end style$
$begin mathsize 12px style integral fraction numerator d x over denominator x squared minus a squared end fraction equals end style$ $begin mathsize 12px style fraction numerator 1 over denominator 2 a end fraction ln open vertical bar fraction numerator x minus a over denominator x plus a end fraction close vertical bar plus C end style$
$begin mathsize 12px style integral fraction numerator d x over denominator square root of a squared minus x squared end root end fraction equals end style$ $begin mathsize 12px style a r c s i n x over a plus C equals end style$ $begin mathsize 12px style negative a r c cos x over a plus C end style$
$begin mathsize 12px style integral fraction numerator d x over denominator square root of a squared plus alpha end root end fraction equals end style$ $begin mathsize 12px style ln open vertical bar x plus square root of x squared plus alpha end root close vertical bar plus C end style$
$begin mathsize 12px style integral square root of x squared plus alpha end root d x equals end style$ $begin mathsize 12px style x over 2 square root of a squared plus alpha end root plus alpha over 2 ln open vertical bar x plus square root of x squared plus alpha end root close vertical bar plus C end style$
$begin mathsize 12px style integral square root of a squared minus x squared end root d x equals end style$ $size 12px x over size 12px 2 square root of size 12px a to the power of size 12px 2 size 12px minus size 12px x to the power of size 12px 2 end root size 12px plus size 12px a to the power of size 12px 2 over size 12px 2 size 12px a size 12px r size 12px c size 12px s size 12px i size 12px n size 12px x over size 12px a size 12px plus size 12px C$
$begin mathsize 12px style integral fraction numerator d x over denominator sin x end fraction equals ln open vertical bar t g x over 2 close vertical bar plus C semicolon end style$ $begin mathsize 12px style integral fraction numerator d x over denominator c o s x end fraction equals ln open vertical bar t g fraction numerator x plus begin display style bevelled straight pi over 2 end style over denominator 22 end fraction close vertical bar plus C semicolon end style$

В формулах 14, 15, 16, 19 предполагается, что а > 0. Каждая из формул таблицы справедлива на любом интервале, на котором непрерывна подынтегральная функция.

Вопросы к конспектам

Вычислите:

begin mathsize 12px style integral fraction numerator 4 d x over denominator s i n squared open parentheses begin display style bevelled straight pi over 3 end style minus 2 x close parentheses end fraction end style

Вычислите:

begin mathsize 12px style integral 3 open parentheses 5 x plus 1 close parentheses squared d x end style

Вычислите:

begin mathsize 12px style f open parentheses x close parentheses equals e to the power of 14 x end exponent plus s i n 2 x end style

Вычислите:

begin mathsize 12px style integral open parentheses fraction numerator x to the power of 4 asterisk times 3 x squared plus 5 cube root of x asterisk times 7 x 16 over denominator cube root of x end fraction close parentheses d x end style

Вычислите:

begin mathsize 12px style integral open parentheses fraction numerator 312 x sin x over denominator x end fraction close parentheses d x end style

Вычислите:

begin mathsize 12px style integral open parentheses fraction numerator sin 2 x over denominator 3 sin x end fraction close parentheses d x end style

Вычислите:

begin mathsize 12px style integral open parentheses e to the power of 5 over 3 x plus 4 end exponent plus 7 to the power of 3 x plus 4 end exponent close parentheses d x end style

Вычислите:

begin mathsize 12px style integral s i n to the power of 4 x d x end style

Вычислите:

begin mathsize 12px style integral c o s x over 4 asterisk times s i n fraction numerator 3 x over denominator 4 end fraction d x end style

Вычислите:

begin mathsize 12px style integral open parentheses fraction numerator 1 over denominator fifth root of x cubed end root end fraction plus 1 over x cubed minus fraction numerator 3 over denominator square root of x end fraction plus fraction numerator x over denominator fourth root of x cubed end root end fraction close parentheses d x end style

Последнее изменение: Среда, 1 Февраль 2017, 18:53