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Beliebt Rechnen >

integral of (2x-3)^{-1}(x+1)^{-1}

Lösung

\int (2 x - 3)^{- 1} (x + 1)^{- 1} d x

Lösung

\frac{1}{5} ln ∣ 2 x - 3 ∣ - \frac{1}{5} ln ∣ x + 1 ∣ + C

Schritte zur Lösung

\int (2 x - 3)^{- 1} (x + 1)^{- 1} d x

Vereinfache

(2 x - 3)^{- 1} (x + 1)^{- 1} : \frac{1}{( 2 x - 3 ) ( x + 1 )}

= \int \frac{1}{( 2 x - 3 ) ( x + 1 )} d x

Ermittle den Partialbruch von

\frac{1}{( 2 x - 3 ) ( x + 1 )} : \frac{2}{5 ( 2 x - 3 )} - \frac{1}{5 ( x + 1 )}

= \int \frac{2}{5 ( 2 x - 3 )} - \frac{1}{5 ( x + 1 )} 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{2}{5 ( 2 x - 3 )} d x - \int \frac{1}{5 ( x + 1 )} d x

\int \frac{2}{5 ( 2 x - 3 )} d x = \frac{1}{5} ln ∣ 2 x - 3 ∣

\int \frac{1}{5 ( x + 1 )} d x = \frac{1}{5} ln ∣ x + 1 ∣

= \frac{1}{5} ln ∣ 2 x - 3 ∣ - \frac{1}{5} ln ∣ x + 1 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{5} ln ∣ 2 x - 3 ∣ - \frac{1}{5} ln ∣ x + 1 ∣ + C

Graph

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