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Beliebt Rechnen >

integral of (2x^3-1)/(x+x^4)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{2 u - 1}{3 u ^{\frac{2}{3}} ( 3 u + u ^{\frac{4}{3}} )} d u

Entferne die Konstante:

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

= \frac{1}{3} \cdot \int \frac{2 u - 1}{u ^{\frac{2}{3}} ( 3 u + u ^{\frac{4}{3}} )} d u

Multipliziere aus

u^{\frac{2}{3}} (3 u + u^{\frac{4}{3}}) : u + u^{2}

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

Multipliziere aus

\frac{2 u - 1}{u + u ^{2}} : \frac{2 u}{u + u ^{2}} - \frac{1}{u + u ^{2}}

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

\int \frac{2 u}{u + u ^{2}} d u = 2 ln ∣ 1 + u ∣

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

= \frac{1}{3} (2 ln ∣ 1 + u ∣ - (ln ∣ u ∣ - ln ∣ u + 1 ∣))

Setze in

u = x^{3}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

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\int ln (21 t) d t

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t an g e n t f (x) = 2 x^{3} + 3 x^{2} - 12 x + 4, at x = 0

(\partial)/(\partial x)(1/((x-y)))

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\int e^{x + 9 e^{x}} d x