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Beliebt Rechnen >

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

Lösung

integral

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

Lösung

6 (- \frac{1}{8} ln - 16 x^{2} + 8 x + 8 + \frac{1}{24} (ln \frac{2}{3} + \frac{4}{3} x - ln - \frac{4}{3} + \frac{4}{3} x)) + C

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Vervollständige das Quadrat

1 + x - 2 x^{2} : - 2 (x - \frac{1}{4})^{2} + \frac{9}{8}

= 3 \cdot \int \frac{x}{- 2 ( x - \frac{1}{4} ) ^{2} + \frac{9}{8}} d x

Wende U-Substitution an

= 3 \cdot \int \frac{2 ( 4 u + 1 )}{- 16 u ^{2} + 9} d u

Entferne die Konstante:

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

= 3 \cdot 2 \cdot \int \frac{4 u + 1}{- 16 u ^{2} + 9} d u

Multipliziere aus

\frac{4 u + 1}{- 16 u ^{2} + 9} : \frac{4 u}{- 16 u ^{2} + 9} + \frac{1}{- 16 u ^{2} + 9}

Wende die Summenregel an:

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

= 3 \cdot 2 (\int \frac{4 u}{- 16 u ^{2} + 9} d u + \int \frac{1}{- 16 u ^{2} + 9} d u)

\int \frac{4 u}{- 16 u ^{2} + 9} d u = - \frac{1}{8} ln - 16 u^{2} + 9

\int \frac{1}{- 16 u ^{2} + 9} d u = \frac{1}{24} (ln \frac{4}{3} u + 1 - ln \frac{4}{3} u - 1)

= 3 \cdot 2 (- \frac{1}{8} ln - 16 u^{2} + 9 + \frac{1}{24} (ln \frac{4}{3} u + 1 - ln \frac{4}{3} u - 1))

Setze in

u = x - \frac{1}{4}

ein

= 3 \cdot 2 (- \frac{1}{8} ln - 16 (x - \frac{1}{4})^{2} + 9 + \frac{1}{24} (ln \frac{4}{3} (x - \frac{1}{4}) + 1 - ln \frac{4}{3} (x - \frac{1}{4}) - 1))

Vereinfache

3 \cdot 2 (- \frac{1}{8} ln - 16 (x - \frac{1}{4})^{2} + 9 + \frac{1}{24} (ln \frac{4}{3} (x - \frac{1}{4}) + 1 - ln \frac{4}{3} (x - \frac{1}{4}) - 1)) : 6 (- \frac{1}{8} ln - 16 x^{2} + 8 x + 8 + \frac{1}{24} (ln \frac{2}{3} + \frac{4}{3} x - ln - \frac{4}{3} + \frac{4}{3} x))

= 6 (- \frac{1}{8} ln - 16 x^{2} + 8 x + 8 + \frac{1}{24} (ln \frac{2}{3} + \frac{4}{3} x - ln - \frac{4}{3} + \frac{4}{3} x))

Füge eine Konstante zur Lösung hinzu

= 6 (- \frac{1}{8} ln - 16 x^{2} + 8 x + 8 + \frac{1}{24} (ln \frac{2}{3} + \frac{4}{3} x - ln - \frac{4}{3} + \frac{4}{3} x)) + C

Graph

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