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Beliebt Rechnen >

integral of (2x^4+3x)/(2x^2)

Lösung

\int \frac{2 x ^{4} + 3 x}{2 x ^{2}} d x

Lösung

\frac{x ^{3}}{3} + \frac{3}{2} ln ∣ x ∣ + C

Schritte zur Lösung

\int \frac{2 x ^{4} + 3 x}{2 x ^{2}} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{2 x ^{4} + 3 x}{x ^{2}} d x

Vereinfache

\frac{2 x ^{4} + 3 x}{x ^{2}} : \frac{2 x ^{3} + 3}{x}

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

Multipliziere aus

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

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

Wende die Summenregel an:

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

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

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

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

Vereinfache

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

= \frac{x ^{3}}{3} + \frac{3}{2} ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{3}}{3} + \frac{3}{2} ln ∣ x ∣ + C

Graph

Beliebte Beispiele

integral of 1/(1+t)

\int \frac{1}{1 + t} d t

integral of e^{x^2}4x

\int e^{x^{2}} 4 x d x

integral from 1 to 3 of x^2

\int_{1}^{3} x^{2} d x

limit as x approaches 4+of (-3)/(8-2x)

x \to 4 + lim (\frac{- 3}{8 - 2 x})

derivative of 6x^{1.5}-4x^{0.5}

\frac{d}{d x} (6 x^{1.5} - 4 x^{0.5})