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Beliebt Rechnen >

integral of (x^2)/(a^2-x^2)

Lösung

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

Lösung

\frac{a}{2} ln \frac{x}{a} + 1 - \frac{a}{2} ln \frac{x}{a} - 1 - x + C

Schritte zur Lösung

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

Wende integrale Substitution an

= \int \frac{a u ^{2}}{1 - u ^{2}} d u

Entferne die Konstante:

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

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

\frac{u ^{2}}{1 - u ^{2}} = \frac{1}{1 - u ^{2}} - 1

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

Wende die Summenregel an:

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

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

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

\int 1 d u = u

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

Setze in

u = \frac{x}{a}

ein

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

Vereinfache

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

= \frac{a}{2} ln \frac{x}{a} + 1 - \frac{a}{2} ln \frac{x}{a} - 1 - x

Füge eine Konstante zur Lösung hinzu

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

Graph

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