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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

d/(dy)((2x^2-xy+2y^2)/(x-y))

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

x^2+2y^2-2x-8y+6=0(0.3)

x^{2} + 2 y^{2} - 2 x - 8 y + 6 = 0 (0.3)

integral of (sqrt(36x^2-49))/(x^3)

\int \frac{36 x ^{2} - 49}{x ^{3}} d x

limit as x approaches pi/2-of sec(x)

x \to \frac{π}{2} - lim (sec (x))

tangent-9x^{1/3}+5,\at x=27

t an g e n t - 9 x^{\frac{1}{3}} + 5, at x = 27