integral of 3/(x^2(x^2+25))

Lösung

\int \frac{3}{x ^{2} ( x ^{2} + 25 )} d x

3 (- \frac{1}{25 x} - \frac{1}{125} arctan (\frac{x}{5})) + C

Schritte zur Lösung

\int \frac{3}{x ^{2} ( x ^{2} + 25 )} 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} ( x ^{2} + 25 )} d x

Ermittle den Partialbruch von

\frac{1}{x ^{2} ( x ^{2} + 25 )} : \frac{1}{25 x ^{2}} - \frac{1}{25 ( x ^{2} + 25 )}

= 3 \cdot \int \frac{1}{25 x ^{2}} - \frac{1}{25 ( x ^{2} + 25 )} 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}{25 x ^{2}} d x - \int \frac{1}{25 ( x ^{2} + 25 )} d x)

\int \frac{1}{25 x ^{2}} d x = - \frac{1}{25 x}

\int \frac{1}{25 ( x ^{2} + 25 )} d x = \frac{1}{125} arctan (\frac{x}{5})

= 3 (- \frac{1}{25 x} - \frac{1}{125} arctan (\frac{x}{5}))

Füge eine Konstante zur Lösung hinzu

= 3 (- \frac{1}{25 x} - \frac{1}{125} arctan (\frac{x}{5})) + C

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