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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

= 2 \cdot \int \frac{1}{2 ( u - 1 ) u ^{2}} d u

Entferne die Konstante:

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

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{( u - 1 ) u ^{2}} d u

Ermittle den Partialbruch von

\frac{1}{( u - 1 ) u ^{2}} : \frac{1}{u - 1} - \frac{1}{u} - \frac{1}{u ^{2}}

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{u - 1} - \frac{1}{u} - \frac{1}{u ^{2}} d u

Wende die Summenregel an:

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

= 2 \cdot \frac{1}{2} (\int \frac{1}{u - 1} d u - \int \frac{1}{u} d u - \int \frac{1}{u ^{2}} d u)

\int \frac{1}{u - 1} d u = ln ∣ u - 1 ∣

\int \frac{1}{u} d u = ln ∣ u ∣

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

= 2 \cdot \frac{1}{2} (ln ∣ u - 1 ∣ - ln ∣ u ∣ - (- \frac{1}{u}))

Setze in

u = x^{2} + 1

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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