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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of x/(\sqrt[4]{x+9)}

\int \frac{x}{4 x + 9} d x

derivative of x^3+2x^2-x-1

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

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

\int_{1}^{2} \frac{x ^{2} + 1}{x ^{2}} d x

derivative x\sqrt[3]{x}

d er i v a t i v e x 3 x

integral from 0 to 4 of x^2-4x

\int_{0}^{4} x^{2} - 4 x d x