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Beliebt Rechnen >

integral of (x+1)/(x^2+10x+18)

Lösung

\int \frac{x + 1}{x ^{2} + 10 x + 18} d x

Lösung

\frac{4 + 7}{2 7} ln x + 5 + 7 + \frac{7 - 4}{2 7} ln x + 5 - 7 + C

Schritte zur Lösung

\int \frac{x + 1}{x ^{2} + 10 x + 18} d x

Vervollständige das Quadrat

x^{2} + 10 x + 18 : (x + 5)^{2} - 7

= \int \frac{x + 1}{( x + 5 ) ^{2} - 7} d x

Wende U-Substitution an

= \int \frac{u - 4}{u ^{2} - 7} d u

Ermittle den Partialbruch von

\frac{u - 4}{u ^{2} - 7} : \frac{4 + 7}{2 7 ( u + 7 )} + \frac{7 - 4}{2 7 ( u - 7 )}

= \int \frac{4 + 7}{2 7 ( u + 7 )} + \frac{7 - 4}{2 7 ( u - 7 )} 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{4 + 7}{2 7 ( u + 7 )} d u + \int \frac{7 - 4}{2 7 ( u - 7 )} d u

\int \frac{4 + 7}{2 7 ( u + 7 )} d u = \frac{4 + 7}{2 7} ln u + 7

\int \frac{7 - 4}{2 7 ( u - 7 )} d u = \frac{7 - 4}{2 7} ln u - 7

= \frac{4 + 7}{2 7} ln u + 7 + \frac{7 - 4}{2 7} ln u - 7

Setze in

u = x + 5

ein

= \frac{4 + 7}{2 7} ln x + 5 + 7 + \frac{7 - 4}{2 7} ln x + 5 - 7

Füge eine Konstante zur Lösung hinzu

= \frac{4 + 7}{2 7} ln x + 5 + 7 + \frac{7 - 4}{2 7} ln x + 5 - 7 + C

Graph

Beliebte Beispiele

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