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Beliebt Rechnen >

integral of (2x)/(x^2-3x-2)

Lösung

\int \frac{2 x}{x ^{2} - 3 x - 2} d x

Lösung

ln 4 x^{2} - 12 x - 8 - \frac{3}{17} (ln \frac{2 x - 3}{17} + 1 - ln \frac{2 x - 3}{17} - 1) + C

Schritte zur Lösung

\int \frac{2 x}{x ^{2} - 3 x - 2} d x

Entferne die Konstante:

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

= 2 \cdot \int \frac{x}{x ^{2} - 3 x - 2} d x

Vervollständige das Quadrat

x^{2} - 3 x - 2 : (x - \frac{3}{2})^{2} - \frac{17}{4}

= 2 \cdot \int \frac{x}{( x - \frac{3}{2} ) ^{2} - \frac{17}{4}} d x

Wende U-Substitution an

= 2 \cdot \int \frac{2 ( 2 u + 3 )}{4 u ^{2} - 17} d u

Entferne die Konstante:

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

= 2 \cdot 2 \cdot \int \frac{2 u + 3}{4 u ^{2} - 17} d u

Multipliziere aus

\frac{2 u + 3}{4 u ^{2} - 17} : \frac{2 u}{4 u ^{2} - 17} + \frac{3}{4 u ^{2} - 17}

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 \cdot 2 (\int \frac{2 u}{4 u ^{2} - 17} d u + \int \frac{3}{4 u ^{2} - 17} d u)

\int \frac{2 u}{4 u ^{2} - 17} d u = \frac{1}{4} ln 4 u^{2} - 17

\int \frac{3}{4 u ^{2} - 17} d u = - \frac{3}{4 17} (ln \frac{2}{17} u + 1 - ln \frac{2}{17} u - 1)

= 2 \cdot 2 (\frac{1}{4} ln 4 u^{2} - 17 - \frac{3}{4 17} (ln \frac{2}{17} u + 1 - ln \frac{2}{17} u - 1))

Setze in

u = x - \frac{3}{2}

ein

= 2 \cdot 2 (\frac{1}{4} ln 4 (x - \frac{3}{2})^{2} - 17 - \frac{3}{4 17} (ln \frac{2}{17} (x - \frac{3}{2}) + 1 - ln \frac{2}{17} (x - \frac{3}{2}) - 1))

Vereinfache

2 \cdot 2 (\frac{1}{4} ln 4 (x - \frac{3}{2})^{2} - 17 - \frac{3}{4 17} (ln \frac{2}{17} (x - \frac{3}{2}) + 1 - ln \frac{2}{17} (x - \frac{3}{2}) - 1)) : ln 4 x^{2} - 12 x - 8 - \frac{3}{17} (ln \frac{2 x - 3}{17} + 1 - ln \frac{2 x - 3}{17} - 1)

= ln 4 x^{2} - 12 x - 8 - \frac{3}{17} (ln \frac{2 x - 3}{17} + 1 - ln \frac{2 x - 3}{17} - 1)

Füge eine Konstante zur Lösung hinzu

= ln 4 x^{2} - 12 x - 8 - \frac{3}{17} (ln \frac{2 x - 3}{17} + 1 - ln \frac{2 x - 3}{17} - 1) + C

Graph

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