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Beliebt Rechnen >

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

Lösung

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

Lösung

2 ln ∣ x ∣ - ln x^{2} + 1 + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{x}{x ^{2} + 1} d x = \frac{1}{2} ln x^{2} + 1

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

Vereinfache

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

= 2 ln ∣ x ∣ - ln x^{2} + 1

Füge eine Konstante zur Lösung hinzu

= 2 ln ∣ x ∣ - ln x^{2} + 1 + C

Graph

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\int y x^{2} d y

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\int \frac{x ^{2} + x - 5}{2 x ^{3} + 3 x ^{2} - 2 x} d x

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\int cos (ln (2 x)) d x

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