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Beliebt Rechnen >

integral of ((4x+7))/(x^2-5x+8)

Lösung

\int \frac{( 4 x + 7 )}{x ^{2} - 5 x + 8} d x

Lösung

4 (\frac{1}{2} ln 4 x^{2} - 20 x + 32 + \frac{17}{2 7} arctan (\frac{2 x - 5}{7})) + C

Schritte zur Lösung

\int \frac{4 x + 7}{x ^{2} - 5 x + 8} d x

Vervollständige das Quadrat

x^{2} - 5 x + 8 : (x - \frac{5}{2})^{2} + \frac{7}{4}

= \int \frac{4 x + 7}{( x - \frac{5}{2} ) ^{2} + \frac{7}{4}} d x

Wende U-Substitution an

= \int \frac{4 ( 4 u + 17 )}{4 u ^{2} + 7} d u

Entferne die Konstante:

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

= 4 \cdot \int \frac{4 u + 17}{4 u ^{2} + 7} d u

Multipliziere aus

\frac{4 u + 17}{4 u ^{2} + 7} : \frac{4 u}{4 u ^{2} + 7} + \frac{17}{4 u ^{2} + 7}

Wende die Summenregel an:

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

= 4 (\int \frac{4 u}{4 u ^{2} + 7} d u + \int \frac{17}{4 u ^{2} + 7} d u)

\int \frac{4 u}{4 u ^{2} + 7} d u = \frac{1}{2} ln 4 u^{2} + 7

\int \frac{17}{4 u ^{2} + 7} d u = \frac{17}{2 7} arctan (\frac{2}{7} u)

= 4 (\frac{1}{2} ln 4 u^{2} + 7 + \frac{17}{2 7} arctan (\frac{2}{7} u))

Setze in

u = x - \frac{5}{2}

ein

= 4 (\frac{1}{2} ln 4 (x - \frac{5}{2})^{2} + 7 + \frac{17}{2 7} arctan (\frac{2}{7} (x - \frac{5}{2})))

Vereinfache

4 (\frac{1}{2} ln 4 (x - \frac{5}{2})^{2} + 7 + \frac{17}{2 7} arctan (\frac{2}{7} (x - \frac{5}{2}))) : 4 (\frac{1}{2} ln 4 x^{2} - 20 x + 32 + \frac{17}{2 7} arctan (\frac{2 x - 5}{7}))

= 4 (\frac{1}{2} ln 4 x^{2} - 20 x + 32 + \frac{17}{2 7} arctan (\frac{2 x - 5}{7}))

Füge eine Konstante zur Lösung hinzu

= 4 (\frac{1}{2} ln 4 x^{2} - 20 x + 32 + \frac{17}{2 7} arctan (\frac{2 x - 5}{7})) + C

Graph

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