integral from 1 to 2 of 1/(sqrt(2x-x^2))

Lösung

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

\frac{π}{2}

Dezimale

1.57079 \dots

Schritte zur Lösung

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

Vervollständige das Quadrat

2 x - x^{2} : - (x - 1)^{2} + 1

= \int_{1}^{2} \frac{1}{- ( x - 1 ) ^{2} + 1} d x

Wende U-Substitution an

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

Nutze das gemeinsame Integral :

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

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

Berechne die Grenzen:

\frac{π}{2}

= \frac{π}{2}

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