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Beliebt Rechnen >

integral of (sqrt(x^2+5))/x

Lösung

\int \frac{x ^{2} + 5}{x} d x

Lösung

- 5 (\frac{1}{2} ln \frac{x ^{2} + 5}{5} + 1 - \frac{1}{2} ln \frac{x ^{2} + 5}{5} - 1 - \frac{1}{5} (x^{2} + 5)) + C

Schritte zur Lösung

\int \frac{x ^{2} + 5}{x} d x

Wende U-Substitution an

= \int \frac{u ^{2}}{u ^{2} - 5} d u

Faktorisiere

u^{2} - 5 : - (- u^{2} + 5)

= \int \frac{u ^{2}}{- ( - u ^{2} + 5 )} 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} + 5} d u

Wende integrale Substitution an

= - \int \frac{5 v ^{2}}{- v ^{2} + 1} d v

Entferne die Konstante:

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

= - 5 \cdot \int \frac{v ^{2}}{- v ^{2} + 1} d v

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

= - 5 \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

= - 5 (\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

= - 5 (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2} - v)

Ersetze zurück

= - 5 \frac{ln \frac{x ^{2} + 5}{5} + 1}{2} - \frac{ln \frac{x ^{2} + 5}{5} - 1}{2} - \frac{x ^{2} + 5}{5}

Vereinfache

- 5 \frac{ln \frac{x ^{2} + 5}{5} + 1}{2} - \frac{ln \frac{x ^{2} + 5}{5} - 1}{2} - \frac{x ^{2} + 5}{5} : - 5 (\frac{1}{2} ln \frac{x ^{2} + 5}{5} + 1 - \frac{1}{2} ln \frac{x ^{2} + 5}{5} - 1 - \frac{1}{5} (x^{2} + 5))

= - 5 (\frac{1}{2} ln \frac{x ^{2} + 5}{5} + 1 - \frac{1}{2} ln \frac{x ^{2} + 5}{5} - 1 - \frac{1}{5} (x^{2} + 5))

Füge eine Konstante zur Lösung hinzu

= - 5 (\frac{1}{2} ln \frac{x ^{2} + 5}{5} + 1 - \frac{1}{2} ln \frac{x ^{2} + 5}{5} - 1 - \frac{1}{5} (x^{2} + 5)) + C

Graph

Beliebte Beispiele

derivative (x^2)/(5+x)

d er i v a t i v e \frac{x ^{2}}{5 + x}

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d er i v a t i v e 10 x

tangent y=(14)/(1+e^{-x)},(0,7)

t an g e n t y = \frac{14}{1 + e ^{- x}}, (0, 7)

integral of 13ln(x^2-x+6)

\int 13 ln (x^{2} - x + 6) d x

derivative of x^4+3

\frac{d}{d x} (x^{4} + 3)