integral from 0 to 3 of 1/((3x+4)^{1/3)}

Lösung

\int_{0}^{3} \frac{1}{( 3 x + 4 ) ^{\frac{1}{3}}} d x

2 - 2^{\frac{1}{3}} + \frac{1 3 ^{\frac{2}{3}} - 4}{2}

Dezimale

1.50446 \dots

Schritte zur Lösung

\int_{0}^{3} \frac{1}{( 3 x + 4 ) ^{\frac{1}{3}}} d x

Multipliziere mit der Konjugation von

3 x + 4 : \frac{9 x ^{2} - 16}{3 x - 4}

= \int_{0}^{3} \frac{1}{( \frac{9 x ^{2} - 16}{3 x - 4} ) ^{\frac{1}{3}}} d x

Bestimme unbestimmte (Singularitäts-)Punkte:

x = \frac{4}{3}

Wenn

b, a < b < c, f (b) =

unbestimmt

, \int_{a}^{c} f (x) d x = \int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x

= \int_{0}^{\frac{4}{3}} \frac{1}{( \frac{9 x ^{2} - 16}{3 x - 4} ) ^{\frac{1}{3}}} d x + \int_{\frac{4}{3}}^{3} \frac{1}{( \frac{9 x ^{2} - 16}{3 x - 4} ) ^{\frac{1}{3}}} d x

\int_{0}^{\frac{4}{3}} \frac{1}{( \frac{9 x ^{2} - 16}{3 x - 4} ) ^{\frac{1}{3}}} d x = 2 - 2^{\frac{1}{3}}

\int_{\frac{4}{3}}^{3} \frac{1}{( \frac{9 x ^{2} - 16}{3 x - 4} ) ^{\frac{1}{3}}} d x = \frac{1 3 ^{\frac{2}{3}} - 4}{2}

= 2 - 2^{\frac{1}{3}} + \frac{1 3 ^{\frac{2}{3}} - 4}{2}

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