integral from 0 to 1 of sqrt(1+9x^{1/2)}

Lösung

\int_{0}^{1} 1 + 9 x^{\frac{1}{2}} d x

\frac{8 + 1000 10}{1215}

Dezimale

2.60928 \dots

Schritte zur Lösung

\int_{0}^{1} 1 + 9 x^{\frac{1}{2}} d x

Multipliziere mit der Konjugation von

1 + 9 x^{\frac{1}{2}} : \frac{1 - 81 x}{1 - 9 x ^{\frac{1}{2}}}

= \int_{0}^{1} \frac{1 - 81 x}{1 - 9 x ^{\frac{1}{2}}} d x

Bestimme unbestimmte (Singularitäts-)Punkte:

x = \frac{1}{81}

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{1}{81}} \frac{1 - 81 x}{1 - 9 x ^{\frac{1}{2}}} d x + \int_{\frac{1}{81}}^{1} \frac{1 - 81 x}{1 - 9 x ^{\frac{1}{2}}} d x

\int_{0}^{\frac{1}{81}} \frac{1 - 81 x}{1 - 9 x ^{\frac{1}{2}}} d x = \frac{2 ( 24 2 + 4 - 20 2 )}{1215}

\int_{\frac{1}{81}}^{1} \frac{1 - 81 x}{1 - 9 x ^{\frac{1}{2}}} d x = \frac{2 ( 600 10 - 24 2 - 5 ( 20 10 - 4 2 ) )}{1215}

= \frac{2 ( 24 2 + 4 - 20 2 )}{1215} + \frac{2 ( 600 10 - 24 2 - 5 ( 20 10 - 4 2 ) )}{1215}

Vereinfache

\frac{2 ( 24 2 + 4 - 20 2 )}{1215} + \frac{2 ( 600 10 - 24 2 - 5 ( 20 10 - 4 2 ) )}{1215} : \frac{8 + 1000 10}{1215}

= \frac{8 + 1000 10}{1215}

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