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Beliebt Rechnen >

integral of (x^2)/(x^2+4)

Lösung

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

Lösung

- 2 arctan (\frac{x}{2}) + x + C

Schritte zur Lösung

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

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

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

Wende integrale Substitution an

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

Entferne die Konstante:

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

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

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

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

\int 1 d u = u

= 2 (- arctan (u) + u)

Setze in

u = \frac{x}{2}

ein

= 2 (- arctan (\frac{x}{2}) + \frac{x}{2})

Vereinfache

2 (- arctan (\frac{x}{2}) + \frac{x}{2}) : - 2 arctan (\frac{x}{2}) + x

= - 2 arctan (\frac{x}{2}) + x

Füge eine Konstante zur Lösung hinzu

= - 2 arctan (\frac{x}{2}) + x + C

Graph

Beliebte Beispiele

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