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

Lösung

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

\frac{1}{2} ln (2) + \frac{π}{8}

Dezimale

0.73927 \dots

Schritte zur Lösung

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

Vervollständige das Quadrat

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

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

Wende U-Substitution an

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

Multipliziere aus

\frac{u + 1}{u ^{2} + 4} : \frac{u}{u ^{2} + 4} + \frac{1}{u ^{2} + 4}

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int_{0}^{2} \frac{u}{u ^{2} + 4} d u = \frac{1}{2} ln (2)

\int_{0}^{2} \frac{1}{u ^{2} + 4} d u = \frac{π}{8}

= \frac{1}{2} ln (2) + \frac{π}{8}

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