integral from 0 to 4 of sqrt(1+x^{1/2)}

Lösung

\int_{0}^{4} 1 + x^{\frac{1}{2}} d x

- 43 + \frac{8 + 108 3}{15}

Dezimale

6.07589 \dots

Schritte zur Lösung

\int_{0}^{4} 1 + x^{\frac{1}{2}} d x

Multipliziere mit der Konjugation von

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

= \int_{0}^{4} \frac{1 - x}{1 - x ^{\frac{1}{2}}} d x

Bestimme unbestimmte (Singularitäts-)Punkte:

x = 1

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}^{1} \frac{1 - x}{1 - x ^{\frac{1}{2}}} d x + \int_{1}^{4} \frac{1 - x}{1 - x ^{\frac{1}{2}}} d x

\int_{0}^{1} \frac{1 - x}{1 - x ^{\frac{1}{2}}} d x = 2 \cdot \frac{24 2 + 4 - 20 2}{15}

\int_{1}^{4} \frac{1 - x}{1 - x ^{\frac{1}{2}}} d x = 2 (\frac{4 2}{3} + \frac{18 3 - 8 2}{5} - 23)

= 2 \cdot \frac{24 2 + 4 - 20 2}{15} + 2 (\frac{4 2}{3} + \frac{18 3 - 8 2}{5} - 23)

Vereinfache

2 \cdot \frac{24 2 + 4 - 20 2}{15} + 2 (\frac{4 2}{3} + \frac{18 3 - 8 2}{5} - 23) : - 43 + \frac{8 + 108 3}{15}

= - 43 + \frac{8 + 108 3}{15}

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