integral from 6 to 7 of x/(x^2+6x+13)

Lösung

\int_{6}^{7} \frac{x}{x ^{2} + 6 x + 13} d x

\frac{ln ( 104 ) - ln ( 85 ) - 3 ( arctan ( 5 ) - arctan ( \frac{9}{2} ) )}{2}

Dezimale

0.06895 \dots

Schritte zur Lösung

\int_{6}^{7} \frac{x}{x ^{2} + 6 x + 13} d x

Vervollständige das Quadrat

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

= \int_{6}^{7} \frac{x}{( x + 3 ) ^{2} + 4} d x

Wende U-Substitution an

= \int_{9}^{10} \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_{9}^{10} \frac{u}{u ^{2} + 4} d u - \int_{9}^{10} \frac{3}{u ^{2} + 4} d u

\int_{9}^{10} \frac{u}{u ^{2} + 4} d u = \frac{1}{2} (ln (104) - ln (85))

\int_{9}^{10} \frac{3}{u ^{2} + 4} d u = \frac{3}{2} (arctan (5) - arctan (\frac{9}{2}))

= \frac{1}{2} (ln (104) - ln (85)) - \frac{3}{2} (arctan (5) - arctan (\frac{9}{2}))

Vereinfache

\frac{1}{2} (ln (104) - ln (85)) - \frac{3}{2} (arctan (5) - arctan (\frac{9}{2})) : \frac{ln ( 104 ) - ln ( 85 ) - 3 ( arctan ( 5 ) - arctan ( \frac{9}{2} ) )}{2}

= \frac{ln ( 104 ) - ln ( 85 ) - 3 ( arctan ( 5 ) - arctan ( \frac{9}{2} ) )}{2}

y^{^{''}} + \frac{1}{2} y^{^{'}} + y = 0

d er i v a t i v e x^{2} + 3 x - 4

t an g e n t x^{2} - 3 x

x \to 0 lim ((1 + 8 x)^{\frac{7}{x}})

y^{^{'}} = 100 k y - k y^{2}