English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of x^2sqrt(x^2-81)

Lösung

\int x^{2} x^{2} - 81 d x

6561 (\frac{1}{19683} x^{3} x^{2} - 81 - \frac{1}{78732} x^{3} x^{2} - 81 - \frac{1}{648} x x^{2} - 81 - \frac{1}{8} ln (\frac{1}{9} x^{2} - 81 + x)) + C

Schritte zur Lösung

\int x^{2} x^{2} - 81 d x

Multipliziere mit der Konjugation von

x^{2} - 81 : \frac{x ^{4} - 6561}{x ^{2} + 81}

= \int x^{2} \frac{x ^{4} - 6561}{x ^{2} + 81} d x

Wende integrale Substitution an

= \int 6561 u^{2} (u + 1) (u - 1) d u

Entferne die Konstante:

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

= 6561 \cdot \int u^{2} (u + 1) (u - 1) d u

Multipliziere mit der Konjugation von

u + 1 : \frac{u ^{2} - 1}{u - 1}

= 6561 \cdot \int u^{2} \frac{u ^{2} - 1}{u - 1} (u - 1) d u

Multipliziere

\frac{u ^{2} - 1}{u - 1} (u - 1) : u^{2} - 1

= 6561 \cdot \int u^{2} u^{2} - 1 d u

Wende die partielle Integration an

= 6561 (\frac{1}{3} u^{3} u^{2} - 1 - \int \frac{u ^{4}}{3 u ^{2} - 1} d u)

\int \frac{u ^{4}}{3 u ^{2} - 1} d u = \frac{1}{12} u^{3} u^{2} - 1 + \frac{1}{8} (u u^{2} - 1 + ln u^{2} - 1 + u)

= 6561 (\frac{1}{3} u^{3} u^{2} - 1 - (\frac{1}{12} u^{3} u^{2} - 1 + \frac{1}{8} (u u^{2} - 1 + ln u^{2} - 1 + u)))

Setze in

u = \frac{x}{9}

ein

= 6561 (\frac{1}{3} (\frac{x}{9})^{3} (\frac{x}{9})^{2} - 1 - (\frac{1}{12} (\frac{x}{9})^{3} (\frac{x}{9})^{2} - 1 + \frac{1}{8} (\frac{x}{9} (\frac{x}{9})^{2} - 1 + ln (\frac{x}{9})^{2} - 1 + \frac{x}{9})))

Vereinfache

6561 (\frac{1}{3} (\frac{x}{9})^{3} (\frac{x}{9})^{2} - 1 - (\frac{1}{12} (\frac{x}{9})^{3} (\frac{x}{9})^{2} - 1 + \frac{1}{8} (\frac{x}{9} (\frac{x}{9})^{2} - 1 + ln (\frac{x}{9})^{2} - 1 + \frac{x}{9}))) : 6561 (\frac{1}{19683} x^{3} x^{2} - 81 - \frac{1}{78732} x^{3} x^{2} - 81 - \frac{1}{648} x x^{2} - 81 - \frac{1}{8} ln (\frac{1}{9} x^{2} - 81 + x))

= 6561 (\frac{1}{19683} x^{3} x^{2} - 81 - \frac{1}{78732} x^{3} x^{2} - 81 - \frac{1}{648} x x^{2} - 81 - \frac{1}{8} ln (\frac{1}{9} x^{2} - 81 + x))

Füge eine Konstante zur Lösung hinzu

= 6561 (\frac{1}{19683} x^{3} x^{2} - 81 - \frac{1}{78732} x^{3} x^{2} - 81 - \frac{1}{648} x x^{2} - 81 - \frac{1}{8} ln (\frac{1}{9} x^{2} - 81 + x)) + C

\int \frac{6}{x ( x ^{2} + 25 )} d x

\frac{d}{d x} (x^{2} - x + 4)

\int \frac{e ^{x}}{e ^{x} + 5} d x

\int x^{13} cos (x^{7}) d x

\frac{\partial}{\partial y} (2 x^{2} - x^{4} + y^{2})