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Beliebt Rechnen >

integral of (y^2+12y+36)/((y^2+36)^2)

Lösung

\int \frac{y ^{2} + 12 y + 36}{( y ^{2} + 36 ) ^{2}} d y

Lösung

\frac{1}{6} arctan (\frac{y}{6}) - \frac{6}{y ^{2} + 36} + C

Schritte zur Lösung

\int \frac{y ^{2} + 12 y + 36}{( y ^{2} + 36 ) ^{2}} d y

Ermittle den Partialbruch von

\frac{y ^{2} + 12 y + 36}{( y ^{2} + 36 ) ^{2}} : \frac{1}{y ^{2} + 36} + \frac{12 y}{( y ^{2} + 36 ) ^{2}}

= \int \frac{1}{y ^{2} + 36} + \frac{12 y}{( y ^{2} + 36 ) ^{2}} d y

Wende die Summenregel an:

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

= \int \frac{1}{y ^{2} + 36} d y + \int \frac{12 y}{( y ^{2} + 36 ) ^{2}} d y

\int \frac{1}{y ^{2} + 36} d y = \frac{1}{6} arctan (\frac{y}{6})

\int \frac{12 y}{( y ^{2} + 36 ) ^{2}} d y = - \frac{6}{y ^{2} + 36}

= \frac{1}{6} arctan (\frac{y}{6}) - \frac{6}{y ^{2} + 36}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{6} arctan (\frac{y}{6}) - \frac{6}{y ^{2} + 36} + C

Graph

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