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integral 1/((x^3+a^3))

Lösung

integral

\frac{1}{( x ^{3} + a ^{3} )}

\frac{1}{6 a ^{2}} (2 ln \frac{x}{a} + 1 + 23 arctan (\frac{2 x - a}{3 a}) - ln \frac{- 4 a x + 4 x ^{2}}{3 a ^{2}} + \frac{4}{3}) + C

Schritte zur Lösung

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

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

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

Wende Exponentenregel an:

\frac{a ^{n}}{b ^{n}} = (\frac{a}{b})^{n}

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

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

Entferne die Konstante:

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

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{3 ( u + 1 )} d u = \frac{1}{3} ln ∣ u + 1 ∣

\int \frac{- u + 2}{3 ( u ^{2} - u + 1 )} d u = \frac{1}{6} (- ln \frac{4}{3} + \frac{4 u ^{2} - 4 u}{3} + 23 arctan (\frac{2 u - 1}{3}))

= \frac{1}{a ^{3}} a (\frac{1}{3} ln ∣ u + 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 u ^{2} - 4 u}{3} + 23 arctan (\frac{2 u - 1}{3})))

Setze in

u = \frac{x}{3 a ^{3}}

ein

= \frac{1}{a ^{3}} a \frac{1}{3} ln \frac{x}{3 a ^{3}} + 1 + \frac{1}{6} - ln \frac{4}{3} + \frac{4 ( \frac{x}{3 a ^{3}} ) ^{2} - 4 \cdot \frac{x}{3 a ^{3}}}{3} + 23 arctan (\frac{2 \cdot \frac{x}{3 a ^{3}} - 1}{3})

Vereinfache

\frac{1}{a ^{3}} a \frac{1}{3} ln \frac{x}{3 a ^{3}} + 1 + \frac{1}{6} - ln \frac{4}{3} + \frac{4 ( \frac{x}{3 a ^{3}} ) ^{2} - 4 \cdot \frac{x}{3 a ^{3}}}{3} + 23 arctan (\frac{2 \cdot \frac{x}{3 a ^{3}} - 1}{3}) : \frac{1}{6 a ^{2}} (2 ln \frac{x}{a} + 1 + 23 arctan (\frac{2 x - a}{3 a}) - ln \frac{- 4 a x + 4 x ^{2}}{3 a ^{2}} + \frac{4}{3})

= \frac{1}{6 a ^{2}} (2 ln \frac{x}{a} + 1 + 23 arctan (\frac{2 x - a}{3 a}) - ln \frac{- 4 a x + 4 x ^{2}}{3 a ^{2}} + \frac{4}{3})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{6 a ^{2}} (2 ln \frac{x}{a} + 1 + 23 arctan (\frac{2 x - a}{3 a}) - ln \frac{- 4 a x + 4 x ^{2}}{3 a ^{2}} + \frac{4}{3}) + C

in t e g r a l 0^{13} x + 1 (x + 1)^{2}

in t e g r a l \frac{e ^{x}}{( x + 2 )}

in t e g r a l e^{x} cos^{2} (x)

in t e g r a l y = cos (x^{2}) (0.1)

in t e g r a l t^{2} (e^{t})^{2}