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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Ermittle den Partialbruch von

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

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

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

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

= - 3 arctan (x) - \frac{x}{1 + x ^{2}} - \frac{2}{x} + C

Graph

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