integral of (y^2+16y+64)/((y^2+64)^2)

Lösung

\int \frac{y ^{2} + 16 y + 64}{( y ^{2} + 64 ) ^{2}} d y

\frac{1}{8} arctan (\frac{y}{8}) - \frac{8}{y ^{2} + 64} + C

Schritte zur Lösung

\int \frac{y ^{2} + 16 y + 64}{( y ^{2} + 64 ) ^{2}} d y

Ermittle den Partialbruch von

\frac{y ^{2} + 16 y + 64}{( y ^{2} + 64 ) ^{2}} : \frac{1}{y ^{2} + 64} + \frac{16 y}{( y ^{2} + 64 ) ^{2}}

= \int \frac{1}{y ^{2} + 64} + \frac{16 y}{( y ^{2} + 64 ) ^{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} + 64} d y + \int \frac{16 y}{( y ^{2} + 64 ) ^{2}} d y

\int \frac{1}{y ^{2} + 64} d y = \frac{1}{8} arctan (\frac{y}{8})

\int \frac{16 y}{( y ^{2} + 64 ) ^{2}} d y = - \frac{8}{y ^{2} + 64}

= \frac{1}{8} arctan (\frac{y}{8}) - \frac{8}{y ^{2} + 64}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{8} arctan (\frac{y}{8}) - \frac{8}{y ^{2} + 64} + C

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