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Beliebt Rechnen >

integral of-(3x-1)/((x-5)(x-7)^2)

Lösung

\int - \frac{3 x - 1}{( x - 5 ) ( x - 7 ) ^{2}} d x

Lösung

- \frac{7}{2} ln ∣ x - 5 ∣ + \frac{7}{2} ln ∣ x - 7 ∣ + \frac{10}{x - 7} + C

Schritte zur Lösung

\int - \frac{3 x - 1}{( x - 5 ) ( x - 7 ) ^{2}} d x

Entferne die Konstante:

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

= - \int \frac{3 x - 1}{( x - 5 ) ( x - 7 ) ^{2}} d x

Ermittle den Partialbruch von

\frac{3 x - 1}{( x - 5 ) ( x - 7 ) ^{2}} : \frac{7}{2 ( x - 5 )} - \frac{7}{2 ( x - 7 )} + \frac{10}{( x - 7 ) ^{2}}

= - \int \frac{7}{2 ( x - 5 )} - \frac{7}{2 ( x - 7 )} + \frac{10}{( x - 7 ) ^{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{7}{2 ( x - 5 )} d x - \int \frac{7}{2 ( x - 7 )} d x + \int \frac{10}{( x - 7 ) ^{2}} d x)

\int \frac{7}{2 ( x - 5 )} d x = \frac{7}{2} ln ∣ x - 5 ∣

\int \frac{7}{2 ( x - 7 )} d x = \frac{7}{2} ln ∣ x - 7 ∣

\int \frac{10}{( x - 7 ) ^{2}} d x = - \frac{10}{x - 7}

= - (\frac{7}{2} ln ∣ x - 5 ∣ - \frac{7}{2} ln ∣ x - 7 ∣ - \frac{10}{x - 7})

Vereinfache

= - \frac{7}{2} ln ∣ x - 5 ∣ + \frac{7}{2} ln ∣ x - 7 ∣ + \frac{10}{x - 7}

Füge eine Konstante zur Lösung hinzu

= - \frac{7}{2} ln ∣ x - 5 ∣ + \frac{7}{2} ln ∣ x - 7 ∣ + \frac{10}{x - 7} + C

Graph

Beliebte Beispiele

integral from 0 to 2pi of cos^3(x)

\int_{0}^{2 π} cos^{3} (x) d x

y^{^{''}} - 2 y^{^{'}} = x

derivative of 9/(2x+1)

\frac{d}{d x} (\frac{9}{2 x + 1})

integral from 1-3x to 1 of (u^3)/(1+u^2)

\int_{1 - 3 x}^{1} \frac{u ^{3}}{1 + u ^{2}} d u

integral of-x^2+6x-5

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