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Beliebt Rechnen >

integral from 0 to 1 of 8/(sqrt(x)(x+1))

Lösung

\int_{0}^{1} \frac{8}{x ( x + 1 )} d x

Lösung

4 π

+1

Dezimale

12.56637 \dots

Schritte zur Lösung

\int_{0}^{1} \frac{8}{x ( x + 1 )} d x

Entferne die Konstante:

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

= 8 \cdot \int_{0}^{1} \frac{1}{x ( x + 1 )} d x

Wende U-Substitution an

= 8 \cdot \int_{0}^{1} \frac{2}{u ^{2} + 1} d u

Entferne die Konstante:

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

= 8 \cdot 2 \cdot \int_{0}^{1} \frac{1}{u ^{2} + 1} d u

Nutze das gemeinsame Integral :

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

= 8 \cdot 2 [arctan (u)]_{0}^{1}

Vereinfache

= 16 [arctan (u)]_{0}^{1}

Berechne die Grenzen:

\frac{π}{4}

= 16 \cdot \frac{π}{4}

Vereinfache

= 4 π

Graph

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