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Beliebt Rechnen >

integral from 3 to 7 of x/(x^2+6x+13)

Lösung

\int_{3}^{7} \frac{x}{x ^{2} + 6 x + 13} d x

Lösung

\frac{ln ( \frac{13}{5} ) - 3 ( arctan ( 5 ) - arctan ( 3 ) )}{2}

+1

Dezimale

0.29122 \dots

Schritte zur Lösung

\int_{3}^{7} \frac{x}{x ^{2} + 6 x + 13} d x

Vervollständige das Quadrat

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

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

Wende U-Substitution an

= \int_{6}^{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_{6}^{10} \frac{u}{u ^{2} + 4} d u - \int_{6}^{10} \frac{3}{u ^{2} + 4} d u

\int_{6}^{10} \frac{u}{u ^{2} + 4} d u = \frac{1}{2} ln (\frac{13}{5})

\int_{6}^{10} \frac{3}{u ^{2} + 4} d u = \frac{3}{2} (arctan (5) - arctan (3))

= \frac{1}{2} ln (\frac{13}{5}) - \frac{3}{2} (arctan (5) - arctan (3))

Vereinfache

\frac{1}{2} ln (\frac{13}{5}) - \frac{3}{2} (arctan (5) - arctan (3)) : \frac{ln ( \frac{13}{5} ) - 3 ( arctan ( 5 ) - arctan ( 3 ) )}{2}

= \frac{ln ( \frac{13}{5} ) - 3 ( arctan ( 5 ) - arctan ( 3 ) )}{2}

Graph

Beliebte Beispiele

(1+x^2)(dy)/(dx)-2xy=x^2+x^4

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

(\partial)/(\partial x)(20e^{5x}y)

\frac{\partial}{\partial x} (20 e^{5 x} y)

f (x) = ln (x^{2} - 1)

derivative of (x^2-6x/((x-3)^2))

\frac{d}{d x} (\frac{x ^{2} - 6 x}{( x - 3 ) ^{2}})

integral of 3/((x-2)^2)

\int \frac{3}{( x - 2 ) ^{2}} d x