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Beliebt Rechnen >

integral of (x+2)/(x^2+9x-10)

Lösung

\int \frac{x + 2}{x ^{2} + 9 x - 10} d x

Lösung

2 (\frac{1}{4} ln 4 x^{2} + 36 x - 40 + \frac{5}{44} (ln \frac{20}{11} + \frac{2}{11} x - ln - \frac{2}{11} + \frac{2}{11} x)) + C

Schritte zur Lösung

\int \frac{x + 2}{x ^{2} + 9 x - 10} d x

Vervollständige das Quadrat

x^{2} + 9 x - 10 : (x + \frac{9}{2})^{2} - \frac{121}{4}

= \int \frac{x + 2}{( x + \frac{9}{2} ) ^{2} - \frac{121}{4}} d x

Wende U-Substitution an

= \int \frac{2 ( 2 u - 5 )}{4 u ^{2} - 121} d u

Entferne die Konstante:

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

= 2 \cdot \int \frac{2 u - 5}{4 u ^{2} - 121} d u

Multipliziere aus

\frac{2 u - 5}{4 u ^{2} - 121} : \frac{2 u}{4 u ^{2} - 121} - \frac{5}{4 u ^{2} - 121}

Wende die Summenregel an:

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

= 2 (\int \frac{2 u}{4 u ^{2} - 121} d u - \int \frac{5}{4 u ^{2} - 121} d u)

\int \frac{2 u}{4 u ^{2} - 121} d u = \frac{1}{4} ln 4 u^{2} - 121

\int \frac{5}{4 u ^{2} - 121} d u = - \frac{5}{44} (ln \frac{2}{11} u + 1 - ln \frac{2}{11} u - 1)

= 2 (\frac{1}{4} ln 4 u^{2} - 121 - (- \frac{5}{44} (ln \frac{2}{11} u + 1 - ln \frac{2}{11} u - 1)))

Setze in

u = x + \frac{9}{2}

ein

= 2 (\frac{1}{4} ln 4 (x + \frac{9}{2})^{2} - 121 - (- \frac{5}{44} (ln \frac{2}{11} (x + \frac{9}{2}) + 1 - ln \frac{2}{11} (x + \frac{9}{2}) - 1)))

Vereinfache

2 (\frac{1}{4} ln 4 (x + \frac{9}{2})^{2} - 121 - (- \frac{5}{44} (ln \frac{2}{11} (x + \frac{9}{2}) + 1 - ln \frac{2}{11} (x + \frac{9}{2}) - 1))) : 2 (\frac{1}{4} ln 4 x^{2} + 36 x - 40 + \frac{5}{44} (ln \frac{20}{11} + \frac{2}{11} x - ln - \frac{2}{11} + \frac{2}{11} x))

= 2 (\frac{1}{4} ln 4 x^{2} + 36 x - 40 + \frac{5}{44} (ln \frac{20}{11} + \frac{2}{11} x - ln - \frac{2}{11} + \frac{2}{11} x))

Füge eine Konstante zur Lösung hinzu

= 2 (\frac{1}{4} ln 4 x^{2} + 36 x - 40 + \frac{5}{44} (ln \frac{20}{11} + \frac{2}{11} x - ln - \frac{2}{11} + \frac{2}{11} x)) + C

Graph

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