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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{ln ( 85 ) - ln ( 29 ) - 3 ( arctan ( \frac{9}{2} ) - arctan ( \frac{5}{2} ) )}{2}

+1

Dezimale

0.29492 \dots

Schritte zur Lösung

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

Vervollständige das Quadrat

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

= \int_{2}^{6} \frac{x}{( x + 3 ) ^{2} + 4} d x

Wende U-Substitution an

= \int_{5}^{9} \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_{5}^{9} \frac{u}{u ^{2} + 4} d u - \int_{5}^{9} \frac{3}{u ^{2} + 4} d u

\int_{5}^{9} \frac{u}{u ^{2} + 4} d u = \frac{1}{2} (ln (85) - ln (29))

\int_{5}^{9} \frac{3}{u ^{2} + 4} d u = \frac{3}{2} (arctan (\frac{9}{2}) - arctan (\frac{5}{2}))

= \frac{1}{2} (ln (85) - ln (29)) - \frac{3}{2} (arctan (\frac{9}{2}) - arctan (\frac{5}{2}))

Vereinfache

\frac{1}{2} (ln (85) - ln (29)) - \frac{3}{2} (arctan (\frac{9}{2}) - arctan (\frac{5}{2})) : \frac{ln ( 85 ) - ln ( 29 ) - 3 ( arctan ( \frac{9}{2} ) - arctan ( \frac{5}{2} ) )}{2}

= \frac{ln ( 85 ) - ln ( 29 ) - 3 ( arctan ( \frac{9}{2} ) - arctan ( \frac{5}{2} ) )}{2}

Graph

Beliebte Beispiele

integral of-1/(x^2+1)

\int - \frac{1}{x ^{2} + 1} d x

fläche y=x^7,-2,3

a re a y = x^{7}, - 2, 3

limit as t approaches 0 of 8/t-8/(t^2+t)

t \to 0 lim (\frac{8}{t} - \frac{8}{t ^{2} + t})

derivative tan(y)

d er i v a t i v e tan (y)

y^'=(9x)/(y+x^2y),y(0)=-5

y^{^{'}} = \frac{9 x}{y + x ^{2} y}, y (0) = - 5