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Beliebt Rechnen >

integral of (z^2)/(z^2+a^2)

Lösung

\int \frac{z ^{2}}{z ^{2} + a ^{2}} d z

Lösung

- a arctan (\frac{z}{a}) + z + C

Schritte zur Lösung

\int \frac{z ^{2}}{z ^{2} + a ^{2}} d z

Wende integrale Substitution an

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

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

= a \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

= a (- \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

= a (- arctan (u) + u)

Setze in

u = \frac{z}{a}

ein

= a (- arctan (\frac{z}{a}) + \frac{z}{a})

Vereinfache

a (- arctan (\frac{z}{a}) + \frac{z}{a}) : - a arctan (\frac{z}{a}) + z

= - a arctan (\frac{z}{a}) + z

Füge eine Konstante zur Lösung hinzu

= - a arctan (\frac{z}{a}) + z + C

Graph

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