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Beliebt Rechnen >

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

Lösung

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

Lösung

2 arctan (x) - ln ∣ x ∣ + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

Vereinfache

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

= 2 arctan (x) - ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= 2 arctan (x) - ln ∣ x ∣ + C

Graph

Beliebte Beispiele

derivative of 7-5x

\frac{d}{d x} (7 - 5 x)

tangent y=arcsin(x/2),\at x=1

t an g e n t y = arcsin (\frac{x}{2}), at x = 1

limit as x approaches infinity of-(5)

x \to \infty lim (- (5))

3y^'+sqrt(x)y=5sqrt(x),y(9)=4

3 y^{^{'}} + x y = 5 x, y (9) = 4

3e^{4x}y^'=-16 x/(y^2),y(0)=1

3 e^{4 x} y^{^{'}} = - 16 \frac{x}{y ^{2}}, y (0) = 1