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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende Exponentenregel an:

(a^{n})^{m} = a^{n \cdot m}

(x)^{3} = x^{\frac{3}{2}}

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

Multipliziere aus

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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