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Beliebt Rechnen >

integral of sqrt((4-x^2)/(x^2))

Lösung

\int \frac{4 - x ^{2}}{x ^{2}} d x

Lösung

- ln \frac{1}{2} 4 - x^{2} + 1 + ln \frac{1}{2} 4 - x^{2} - 1 + 4 - x^{2} + C

Schritte zur Lösung

\int \frac{4 - x ^{2}}{x ^{2}} d x

Vereinfache

\frac{4 - x ^{2}}{x ^{2}} : \frac{4 - x ^{2}}{x ^{2}}

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

x^{2} = (x),

angenommen

x \geq 0

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

Wende U-Substitution an

= \int - \frac{u ^{2}}{- u ^{2} + 4} 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} + 4} d u

Wende integrale Substitution an

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

Entferne die Konstante:

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

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

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

= - 2 \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

= - 2 (\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

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

Ersetze zurück

= - 2 \frac{ln \frac{4 - x ^{2}}{2} + 1}{2} - \frac{ln \frac{4 - x ^{2}}{2} - 1}{2} - \frac{4 - x ^{2}}{2}

Vereinfache

- 2 \frac{ln \frac{4 - x ^{2}}{2} + 1}{2} - \frac{ln \frac{4 - x ^{2}}{2} - 1}{2} - \frac{4 - x ^{2}}{2} : - ln \frac{1}{2} 4 - x^{2} + 1 + ln \frac{1}{2} 4 - x^{2} - 1 + 4 - x^{2}

= - ln \frac{1}{2} 4 - x^{2} + 1 + ln \frac{1}{2} 4 - x^{2} - 1 + 4 - x^{2}

Füge eine Konstante zur Lösung hinzu

= - ln \frac{1}{2} 4 - x^{2} + 1 + ln \frac{1}{2} 4 - x^{2} - 1 + 4 - x^{2} + C

Graph

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