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Beliebt Rechnen >

integral of 2/((x-6)(x-4))

Lösung

\int \frac{2}{( x - 6 ) ( x - 4 )} d x

Lösung

ln ∣ x - 6 ∣ - ln ∣ x - 4 ∣ + C

Schritte zur Lösung

\int \frac{2}{( x - 6 ) ( x - 4 )} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{1}{( x - 6 ) ( x - 4 )} d x

Ermittle den Partialbruch von

\frac{1}{( x - 6 ) ( x - 4 )} : \frac{1}{2 ( x - 6 )} - \frac{1}{2 ( x - 4 )}

= 2 \cdot \int \frac{1}{2 ( x - 6 )} - \frac{1}{2 ( x - 4 )} 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

= 2 (\int \frac{1}{2 ( x - 6 )} d x - \int \frac{1}{2 ( x - 4 )} d x)

\int \frac{1}{2 ( x - 6 )} d x = \frac{1}{2} ln ∣ x - 6 ∣

\int \frac{1}{2 ( x - 4 )} d x = \frac{1}{2} ln ∣ x - 4 ∣

= 2 (\frac{1}{2} ln ∣ x - 6 ∣ - \frac{1}{2} ln ∣ x - 4 ∣)

Vereinfache

2 (\frac{1}{2} ln ∣ x - 6 ∣ - \frac{1}{2} ln ∣ x - 4 ∣) : ln ∣ x - 6 ∣ - ln ∣ x - 4 ∣

= ln ∣ x - 6 ∣ - ln ∣ x - 4 ∣

Füge eine Konstante zur Lösung hinzu

= ln ∣ x - 6 ∣ - ln ∣ x - 4 ∣ + C

Graph

Beliebte Beispiele

integral of r^2e^{-(2r)/a}

\int r^{2} e^{- \frac{2 r}{a}} d r

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\int 9 \cdot x^{- 6} - 81 \cdot x^{- 8} d x

integral of (((2+x))/x)^2

\int (\frac{( 2 + x )}{x})^{2} d x

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\int sec^{2} (7 θ) d θ

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