integral of 3/((x^2+1)(x^2+4))

Lösung

\int \frac{3}{( x ^{2} + 1 ) ( x ^{2} + 4 )} d x

arctan (x) - \frac{1}{2} arctan (\frac{x}{2}) + C

Schritte zur Lösung

\int \frac{3}{( x ^{2} + 1 ) ( x ^{2} + 4 )} d x

Entferne die Konstante:

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

= 3 \cdot \int \frac{1}{( x ^{2} + 1 ) ( x ^{2} + 4 )} d x

Ermittle den Partialbruch von

\frac{1}{( x ^{2} + 1 ) ( x ^{2} + 4 )} : \frac{1}{3 ( x ^{2} + 1 )} - \frac{1}{3 ( x ^{2} + 4 )}

= 3 \cdot \int \frac{1}{3 ( x ^{2} + 1 )} - \frac{1}{3 ( x ^{2} + 4 )} d x

Wende die Summenregel an:

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

= 3 (\int \frac{1}{3 ( x ^{2} + 1 )} d x - \int \frac{1}{3 ( x ^{2} + 4 )} d x)

\int \frac{1}{3 ( x ^{2} + 1 )} d x = \frac{1}{3} arctan (x)

\int \frac{1}{3 ( x ^{2} + 4 )} d x = \frac{1}{6} arctan (\frac{x}{2})

= 3 (\frac{1}{3} arctan (x) - \frac{1}{6} arctan (\frac{x}{2}))

Vereinfache

3 (\frac{1}{3} arctan (x) - \frac{1}{6} arctan (\frac{x}{2})) : arctan (x) - \frac{1}{2} arctan (\frac{x}{2})

= arctan (x) - \frac{1}{2} arctan (\frac{x}{2})

Füge eine Konstante zur Lösung hinzu

= arctan (x) - \frac{1}{2} arctan (\frac{x}{2}) + C

\int \frac{8 x ^{3}}{5 - x ^{4}} d x

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