integral of (x^2+1)/((x-9)(x-8)^2)

Lösung

\int \frac{x ^{2} + 1}{( x - 9 ) ( x - 8 ) ^{2}} d x

82 ln ∣ x - 9 ∣ - 81 ln ∣ x - 8 ∣ + \frac{65}{x - 8} + C

Schritte zur Lösung

\int \frac{x ^{2} + 1}{( x - 9 ) ( x - 8 ) ^{2}} d x

Ermittle den Partialbruch von

\frac{x ^{2} + 1}{( x - 9 ) ( x - 8 ) ^{2}} : \frac{82}{x - 9} - \frac{81}{x - 8} - \frac{65}{( x - 8 ) ^{2}}

= \int \frac{82}{x - 9} - \frac{81}{x - 8} - \frac{65}{( x - 8 ) ^{2}} d x

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{82}{x - 9} d x - \int \frac{81}{x - 8} d x - \int \frac{65}{( x - 8 ) ^{2}} d x

\int \frac{82}{x - 9} d x = 82 ln ∣ x - 9 ∣

\int \frac{81}{x - 8} d x = 81 ln ∣ x - 8 ∣

\int \frac{65}{( x - 8 ) ^{2}} d x = - \frac{65}{x - 8}

= 82 ln ∣ x - 9 ∣ - 81 ln ∣ x - 8 ∣ - (- \frac{65}{x - 8})

Vereinfache

= 82 ln ∣ x - 9 ∣ - 81 ln ∣ x - 8 ∣ + \frac{65}{x - 8}

Füge eine Konstante zur Lösung hinzu

= 82 ln ∣ x - 9 ∣ - 81 ln ∣ x - 8 ∣ + \frac{65}{x - 8} + C

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