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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{2 x + 3} d x = \frac{1}{2} ln ∣ 2 x + 3 ∣

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

derivative of (x^2/(sqrt(x^4+1)))

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

\frac{d y}{d x} = 2 y - 1

x^2(dy)/(dx)+y^2=xy

x^{2} \frac{d y}{d x} + y^{2} = x y

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\int cot^{2} (x) csc (x) d x

(\partial)/(\partial x)(-e^{2y}cos(2x)*2)

\frac{\partial}{\partial x} (- e^{2 y} cos (2 x) \cdot 2)