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

Lösung

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

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

Schritte zur Lösung

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

Multipliziere aus

\frac{2 x + 1}{x ^{3} + x} : \frac{2 x}{x ^{3} + x} + \frac{1}{x ^{3} + x}

Wende die Summenregel an:

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

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

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

\int \frac{1}{x ^{3} + x} d x = ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1

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

Füge eine Konstante zur Lösung hinzu

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

in t e g r a l x ln (x)

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