интеграл от 1/((x^2+21)^3)

Решение

\int \frac{1}{( x ^{2} + 21 ) ^{3}} d x

\frac{x ^{4} arctan ( \frac{x}{21} ) + 21 x ^{3} + 42 x ^{2} arctan ( \frac{x}{21} ) + 35 21 x + 441 arctan ( \frac{x}{21} )}{1176 21 ( x ^{2} + 21 ) ^{2}} + C

Шаги решения

\int \frac{1}{( x ^{2} + 21 ) ^{3}} d x

Примените правило возведения в степень:

\frac{1}{a ^{b}} = a^{- b}

\frac{1}{( x ^{2} + 21 ) ^{3}} = (x^{2} + 21)^{- 3}

= \int (x^{2} + 21)^{- 3} d x

Применить подстановку интеграла

= \int \frac{21}{( 21 u ^{2} + 21 ) ^{3}} d u

Извлечь константу:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 21 \cdot \int \frac{1}{( 21 u ^{2} + 21 ) ^{3}} d u

коэффициент

\frac{1}{( 21 u ^{2} + 21 ) ^{3}}

= 21 \cdot \int \frac{1}{9261 ( u ^{2} + 1 ) ^{3}} d u

Извлечь константу:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 21 \frac{1}{9261} \cdot \int \frac{1}{( u ^{2} + 1 ) ^{3}} d u

Применить сокращение интеграла

= 21 \frac{1}{9261} (\frac{3}{4} \cdot \int \frac{1}{( u ^{2} + 1 ) ^{2}} d u + \frac{u}{2 \cdot 2 ( u ^{2} + 1 ) ^{2}})

\int \frac{1}{( u ^{2} + 1 ) ^{2}} d u = \frac{1}{2} (arctan (u) + \frac{u}{1 + u ^{2}})

= 21 \frac{1}{9261} (\frac{3}{4} \cdot \frac{1}{2} (arctan (u) + \frac{u}{1 + u ^{2}}) + \frac{u}{2 \cdot 2 ( u ^{2} + 1 ) ^{2}})

Делаем обратную замену

u = \frac{x}{21}

= 21 \frac{1}{9261} \frac{3}{4} \cdot \frac{1}{2} arctan (\frac{x}{21}) + \frac{\frac{x}{21}}{1 + ( \frac{x}{21} ) ^{2}} + \frac{\frac{x}{21}}{2 \cdot 2 ( ( \frac{x}{21} ) ^{2} + 1 ) ^{2}}

Упростите

21 \frac{1}{9261} \frac{3}{4} \cdot \frac{1}{2} arctan (\frac{x}{21}) + \frac{\frac{x}{21}}{1 + ( \frac{x}{21} ) ^{2}} + \frac{\frac{x}{21}}{2 \cdot 2 ( ( \frac{x}{21} ) ^{2} + 1 ) ^{2}} : \frac{x ^{4} arctan ( \frac{x}{21} ) + 21 x ^{3} + 42 x ^{2} arctan ( \frac{x}{21} ) + 35 21 x + 441 arctan ( \frac{x}{21} )}{1176 21 ( x ^{2} + 21 ) ^{2}}

= \frac{x ^{4} arctan ( \frac{x}{21} ) + 21 x ^{3} + 42 x ^{2} arctan ( \frac{x}{21} ) + 35 21 x + 441 arctan ( \frac{x}{21} )}{1176 21 ( x ^{2} + 21 ) ^{2}}

Добавить константу к решению

= \frac{x ^{4} arctan ( \frac{x}{21} ) + 21 x ^{3} + 42 x ^{2} arctan ( \frac{x}{21} ) + 35 21 x + 441 arctan ( \frac{x}{21} )}{1176 21 ( x ^{2} + 21 ) ^{2}} + C