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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (x^2+1)/(sqrt(4-x^2))

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\int 2 x ln (x^{3}) d x

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\int x 2 - 5 x d x