de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of (sqrt(81+x^2))/x

Lösung

\int \frac{81 + x ^{2}}{x} d x

Lösung

- \frac{9}{2} ln \frac{81 + x ^{2}}{9} + 1 + \frac{9}{2} ln \frac{81 + x ^{2}}{9} - 1 + 81 + x^{2} + C

Schritte zur Lösung

\int \frac{81 + x ^{2}}{x} d x

Wende U-Substitution an

= \int \frac{u ^{2}}{u ^{2} - 81} d u

Faktorisiere

u^{2} - 81 : - (- u^{2} + 81)

= \int \frac{u ^{2}}{- ( - u ^{2} + 81 )} d u

Entferne die Konstante:

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

= - \int \frac{u ^{2}}{- u ^{2} + 81} d u

Wende integrale Substitution an

= - \int \frac{9 v ^{2}}{- v ^{2} + 1} d v

Entferne die Konstante:

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

= - 9 \cdot \int \frac{v ^{2}}{- v ^{2} + 1} d v

\frac{v ^{2}}{- v ^{2} + 1} = \frac{1}{- v ^{2} + 1} - 1

= - 9 \cdot \int \frac{1}{- v ^{2} + 1} - 1 d v

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - 9 (\int \frac{1}{- v ^{2} + 1} d v - \int 1 d v)

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

\int 1 d v = v

= - 9 (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2} - v)

Ersetze zurück

= - 9 \frac{ln \frac{81 + x ^{2}}{9} + 1}{2} - \frac{ln \frac{81 + x ^{2}}{9} - 1}{2} - \frac{81 + x ^{2}}{9}

Vereinfache

- 9 \frac{ln \frac{81 + x ^{2}}{9} + 1}{2} - \frac{ln \frac{81 + x ^{2}}{9} - 1}{2} - \frac{81 + x ^{2}}{9} : - \frac{9}{2} ln \frac{81 + x ^{2}}{9} + 1 + \frac{9}{2} ln \frac{81 + x ^{2}}{9} - 1 + 81 + x^{2}

= - \frac{9}{2} ln \frac{81 + x ^{2}}{9} + 1 + \frac{9}{2} ln \frac{81 + x ^{2}}{9} - 1 + 81 + x^{2}

Füge eine Konstante zur Lösung hinzu

= - \frac{9}{2} ln \frac{81 + x ^{2}}{9} + 1 + \frac{9}{2} ln \frac{81 + x ^{2}}{9} - 1 + 81 + x^{2} + C

Graph

Beliebte Beispiele

limit as x approaches infinity of \sqrt[3]{x^3+3x^2-x}

x \to \infty lim (3 x^{3} + 3 x^{2} - x)

integral of sqrt((6-x)/(6+x))

\int \frac{6 - x}{6 + x} d x

sum from n=0 to infinity of (2^n)/(1^n)

n = 0 \sum \infty \frac{2 ^{n}}{1 ^{n}}

integral of 4cos^3(4x)

\int 4 cos^{3} (4 x) d x

limit as x approaches 1 of 1*0

x \to 1 lim (1 \cdot 0)