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Beliebt Rechnen >

integral of (5x-4)/(sqrt(x^2+6x+13))

Lösung

\int \frac{5 x - 4}{x ^{2} + 6 x + 13} d x

Lösung

5 x^{2} + 6 x + 13 - 19 ln (\frac{1}{2} x + 3 + x^{2} + 6 x + 13) + C

Schritte zur Lösung

\int \frac{5 x - 4}{x ^{2} + 6 x + 13} d x

Vervollständige das Quadrat

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

= \int \frac{5 x - 4}{( x + 3 ) ^{2} + 4} d x

Wende U-Substitution an

= \int \frac{5 u - 19}{u ^{2} + 4} d u

Multipliziere aus

\frac{5 u - 19}{u ^{2} + 4} : \frac{5 u}{u ^{2} + 4} - \frac{19}{u ^{2} + 4}

= \int \frac{5 u}{u ^{2} + 4} - \frac{19}{u ^{2} + 4} d u

Wende die Summenregel an:

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

= \int \frac{5 u}{u ^{2} + 4} d u - \int \frac{19}{u ^{2} + 4} d u

\int \frac{5 u}{u ^{2} + 4} d u = 5 u^{2} + 4

\int \frac{19}{u ^{2} + 4} d u = 19 ln (\frac{1}{2} u + 4 + u^{2})

= 5 u^{2} + 4 - 19 ln (\frac{1}{2} u + 4 + u^{2})

Setze in

u = x + 3

ein

= 5 (x + 3)^{2} + 4 - 19 ln (\frac{1}{2} x + 3 + 4 + (x + 3)^{2})

Vereinfache

5 (x + 3)^{2} + 4 - 19 ln (\frac{1}{2} x + 3 + 4 + (x + 3)^{2}) : 5 x^{2} + 6 x + 13 - 19 ln (\frac{1}{2} x + 3 + x^{2} + 6 x + 13)

= 5 x^{2} + 6 x + 13 - 19 ln (\frac{1}{2} x + 3 + x^{2} + 6 x + 13)

Füge eine Konstante zur Lösung hinzu

= 5 x^{2} + 6 x + 13 - 19 ln (\frac{1}{2} x + 3 + x^{2} + 6 x + 13) + C

Graph

Beliebte Beispiele

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x^2y^{''}-2xy^'-4y=0

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\int (\frac{1}{x + 1}) d x

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