integral from 1 to 3 of 1/(x^2-4x+5)

Lösung

\int_{1}^{3} \frac{1}{x ^{2} - 4 x + 5} d x

\frac{π}{2}

Dezimale

1.57079 \dots

Schritte zur Lösung

\int_{1}^{3} \frac{1}{x ^{2} - 4 x + 5} d x

Vervollständige das Quadrat

x^{2} - 4 x + 5 : (x - 2)^{2} + 1

= \int_{1}^{3} \frac{1}{( x - 2 ) ^{2} + 1} d x

Wende U-Substitution an

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

Nutze das gemeinsame Integral :

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

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

Berechne die Grenzen:

\frac{π}{2}

= \frac{π}{2}

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