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Beliebt Rechnen >

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

Lösung

integral

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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