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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{2 x ^{6} - 1}{8 x ^{2}} + C

Schritte zur Lösung

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

Vereinfache

1 + (x^{3} - \frac{1}{4 x ^{3}})^{2} : \frac{8 x ^{6} + 1 + 16 x ^{12}}{4 x ^{3}}

= \int \frac{8 x ^{6} + 1 + 16 x ^{12}}{4 x ^{3}} d x

Entferne die Konstante:

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

= \frac{1}{4} \cdot \int \frac{8 x ^{6} + 1 + 16 x ^{12}}{x ^{3}} d x

Wende U-Substitution an

= \frac{1}{4} \cdot \int \frac{4 u + 1}{6 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}{4} \cdot \frac{1}{6} \cdot \int \frac{4 u + 1}{u ^{\frac{4}{3}}} d u

Multipliziere aus

\frac{4 u + 1}{u ^{\frac{4}{3}}} : \frac{4}{u ^{\frac{1}{3}}} + \frac{1}{u ^{\frac{4}{3}}}

= \frac{1}{4} \cdot \frac{1}{6} \cdot \int \frac{4}{u ^{\frac{1}{3}}} + \frac{1}{u ^{\frac{4}{3}}} d u

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}{4} \cdot \frac{1}{6} (\int \frac{4}{u ^{\frac{1}{3}}} d u + \int \frac{1}{u ^{\frac{4}{3}}} d u)

\int \frac{4}{u ^{\frac{1}{3}}} d u = 6 u^{\frac{2}{3}}

\int \frac{1}{u ^{\frac{4}{3}}} d u = - \frac{3}{3 u}

= \frac{1}{4} \cdot \frac{1}{6} (6 u^{\frac{2}{3}} - \frac{3}{3 u})

Setze in

u = x^{6}

ein

= \frac{1}{4} \cdot \frac{1}{6} (6 (x^{6})^{\frac{2}{3}} - \frac{3}{3 x ^{6}})

Vereinfache

\frac{1}{4} \cdot \frac{1}{6} (6 (x^{6})^{\frac{2}{3}} - \frac{3}{3 x ^{6}}) : \frac{2 x ^{6} - 1}{8 x ^{2}}

= \frac{2 x ^{6} - 1}{8 x ^{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{2 x ^{6} - 1}{8 x ^{2}} + C

Graph

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