integral from x to x^2 of 1/(t^3+2)

Lösung

\int_{x}^{x^{2}} \frac{1}{t ^{3} + 2} d t

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

Schritte zur Lösung

\int_{x}^{x^{2}} \frac{1}{t ^{3} + 2} d t

t^{3} + 2 = 2 (\frac{t ^{3}}{2} + 1)

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

Wende Exponentenregel an:

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

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

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

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int_{x}^{x^{2}} \frac{1}{( \frac{t}{3 2} ) ^{3} + 1} d t

Wende U-Substitution an

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

Entferne die Konstante:

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

= \frac{1}{2} 32 \cdot \int_{\frac{x}{3 2}}^{\frac{x ^{2}}{3 2}} \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}{2} 32 \cdot \int_{\frac{x}{3 2}}^{\frac{x ^{2}}{3 2}} \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}{2} 32 (\int_{\frac{x}{3 2}}^{\frac{x ^{2}}{3 2}} \frac{1}{3 ( u + 1 )} d u + \int_{\frac{x}{3 2}}^{\frac{x ^{2}}{3 2}} \frac{- u + 2}{3 ( u ^{2} - u + 1 )} d u)

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

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

= \frac{1}{2} 32 \frac{1}{3} (ln \frac{x ^{2}}{3 2} + 1 - ln \frac{x}{3 2} + 1) + \frac{- ln \frac{\frac{2 3 2 x ^{4} - 2 ^{1 + \frac{2}{3}} x ^{2} + 1}{3} + 1}{\frac{2 3 2 x ^{2} - 2 ^{1 + \frac{2}{3}} x + 1}{3} + 1} + 2 3 ( arctan ( \frac{2 ^{\frac{2}{3}} x ^{2} - 1}{3} ) - arctan ( \frac{2 ^{\frac{2}{3}} x - 1}{3} ) )}{6}

Vereinfache

\frac{1}{2} 32 \frac{1}{3} (ln \frac{x ^{2}}{3 2} + 1 - ln \frac{x}{3 2} + 1) + \frac{- ln \frac{\frac{2 3 2 x ^{4} - 2 ^{1 + \frac{2}{3}} x ^{2} + 1}{3} + 1}{\frac{2 3 2 x ^{2} - 2 ^{1 + \frac{2}{3}} x + 1}{3} + 1} + 2 3 ( arctan ( \frac{2 ^{\frac{2}{3}} x ^{2} - 1}{3} ) - arctan ( \frac{2 ^{\frac{2}{3}} x - 1}{3} ) )}{6} : \frac{2 ( ln \frac{x ^{2}}{3 2} + 1 - ln \frac{x}{3 2} + 1 ) + 2 3 ( arctan ( \frac{2 ^{\frac{2}{3}} x ^{2} - 1}{3} ) - arctan ( \frac{2 ^{\frac{2}{3}} x - 1}{3} ) ) - ln \frac{\frac{2 3 2 x ^{4} - 2 ^{1 + \frac{2}{3}} x ^{2} + 1}{3} + 1}{\frac{2 3 2 x ^{2} - 2 ^{1 + \frac{2}{3}} x + 1}{3} + 1}}{6 \cdot 2 ^{\frac{2}{3}}}

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

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