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Beliebt Rechnen >

integral from 4 to 8 of x/(x^2+6x+13)

Lösung

\int_{4}^{8} \frac{x}{x ^{2} + 6 x + 13} d x

Lösung

\frac{3 ln ( 5 ) - ln ( 53 ) - 3 ( arctan ( \frac{11}{2} ) - arctan ( \frac{7}{2} ) )}{2}

+1

Dezimale

0.28134 \dots

Schritte zur Lösung

\int_{4}^{8} \frac{x}{x ^{2} + 6 x + 13} d x

Vervollständige das Quadrat

x^{2} + 6 x + 13 : (x + 3)^{2} + 4

= \int_{4}^{8} \frac{x}{( x + 3 ) ^{2} + 4} d x

Wende U-Substitution an

= \int_{7}^{11} \frac{u - 3}{u ^{2} + 4} d u

Multipliziere aus

\frac{u - 3}{u ^{2} + 4} : \frac{u}{u ^{2} + 4} - \frac{3}{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_{7}^{11} \frac{u}{u ^{2} + 4} d u - \int_{7}^{11} \frac{3}{u ^{2} + 4} d u

\int_{7}^{11} \frac{u}{u ^{2} + 4} d u = \frac{1}{2} (3 ln (5) - ln (53))

\int_{7}^{11} \frac{3}{u ^{2} + 4} d u = \frac{3}{2} (arctan (\frac{11}{2}) - arctan (\frac{7}{2}))

= \frac{1}{2} (3 ln (5) - ln (53)) - \frac{3}{2} (arctan (\frac{11}{2}) - arctan (\frac{7}{2}))

Vereinfache

\frac{1}{2} (3 ln (5) - ln (53)) - \frac{3}{2} (arctan (\frac{11}{2}) - arctan (\frac{7}{2})) : \frac{3 ln ( 5 ) - ln ( 53 ) - 3 ( arctan ( \frac{11}{2} ) - arctan ( \frac{7}{2} ) )}{2}

= \frac{3 ln ( 5 ) - ln ( 53 ) - 3 ( arctan ( \frac{11}{2} ) - arctan ( \frac{7}{2} ) )}{2}

Graph

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