integral from 0 to 1 of x^5sqrt(1+4x^2)

Lösung

\int_{0}^{1} x^{5} 1 + 4 x^{2} d x

\frac{- 1 + 125 5}{840}

Dezimale

0.33155 \dots

Schritte zur Lösung

\int_{0}^{1} x^{5} 1 + 4 x^{2} d x

Multipliziere mit der Konjugation von

1 + 4 x^{2} : \frac{1 - 16 x ^{4}}{1 - 4 x ^{2}}

= \int_{0}^{1} x^{5} \frac{1 - 16 x ^{4}}{1 - 4 x ^{2}} d x

Bestimme unbestimmte (Singularitäts-)Punkte:

x = \frac{1}{2}

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}{2}} x^{5} \frac{1 - 16 x ^{4}}{1 - 4 x ^{2}} d x + \int_{\frac{1}{2}}^{1} x^{5} \frac{1 - 16 x ^{4}}{1 - 4 x ^{2}} d x

\int_{0}^{\frac{1}{2}} x^{5} \frac{1 - 16 x ^{4}}{1 - 4 x ^{2}} d x = \frac{11 2 - 4}{3360}

\int_{\frac{1}{2}}^{1} x^{5} \frac{1 - 16 x ^{4}}{1 - 4 x ^{2}} d x = \frac{- 11 2 + 500 5}{3360}

= \frac{11 2 - 4}{3360} + \frac{- 11 2 + 500 5}{3360}

Vereinfache

\frac{11 2 - 4}{3360} + \frac{- 11 2 + 500 5}{3360} : \frac{- 1 + 125 5}{840}

= \frac{- 1 + 125 5}{840}

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