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Beliebt Rechnen >

integral of 1/((x+1)(2x+1)^2)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

tangent x^3-3x^2-9x+11,\at x=3

t an g e n t x^{3} - 3 x^{2} - 9 x + 11, at x = 3

integral of (cos(θ))^2

\int (cos (θ))^{2} d θ

limit as x approaches 6 of sqrt(x^3+3x)

x \to 6 lim (x^{3} + 3 x)

integral of (2+1/x)^2

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

derivative of (x^2+8^4)

\frac{d}{d x} ((x^{2} + 8)^{4})