integral of (5x+31)/(3x^2-4x+11)

Lösung

\int \frac{5 x + 31}{3 x ^{2} - 4 x + 11} d x

\frac{5}{6} ln 9 x^{2} - 12 x + 33 + \frac{10}{3 29} arctan (\frac{3 x - 2}{29}) + \frac{31}{29} arctan (\frac{3 x - 2}{29}) + C

Schritte zur Lösung

\int \frac{5 x + 31}{3 x ^{2} - 4 x + 11} d x

Vervollständige das Quadrat

3 x^{2} - 4 x + 11 : 3 (x - \frac{2}{3})^{2} + \frac{29}{3}

= \int \frac{5 x + 31}{3 ( x - \frac{2}{3} ) ^{2} + \frac{29}{3}} d x

Wende U-Substitution an

= \int \frac{3 ( 5 ( u + \frac{2}{3} ) + 31 )}{9 u ^{2} + 29} d u

Entferne die Konstante:

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

= 3 \cdot \int \frac{5 ( u + \frac{2}{3} ) + 31}{9 u ^{2} + 29} d u

Multipliziere aus

\frac{5 ( u + \frac{2}{3} ) + 31}{9 u ^{2} + 29} : \frac{5 ( u + \frac{2}{3} )}{9 u ^{2} + 29} + \frac{31}{9 u ^{2} + 29}

Wende die Summenregel an:

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

= 3 (\int \frac{5 ( u + \frac{2}{3} )}{9 u ^{2} + 29} d u + \int \frac{31}{9 u ^{2} + 29} d u)

\int \frac{5 ( u + \frac{2}{3} )}{9 u ^{2} + 29} d u = 5 (\frac{1}{18} ln 9 u^{2} + 29 + \frac{2}{9 29} arctan (\frac{3}{29} u))

\int \frac{31}{9 u ^{2} + 29} d u = \frac{31}{3 29} arctan (\frac{3}{29} u)

= 3 (5 (\frac{1}{18} ln 9 u^{2} + 29 + \frac{2}{9 29} arctan (\frac{3}{29} u)) + \frac{31}{3 29} arctan (\frac{3}{29} u))

Setze in

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

ein

= 3 (5 (\frac{1}{18} ln 9 (x - \frac{2}{3})^{2} + 29 + \frac{2}{9 29} arctan (\frac{3}{29} (x - \frac{2}{3}))) + \frac{31}{3 29} arctan (\frac{3}{29} (x - \frac{2}{3})))

Vereinfache

3 (5 (\frac{1}{18} ln 9 (x - \frac{2}{3})^{2} + 29 + \frac{2}{9 29} arctan (\frac{3}{29} (x - \frac{2}{3}))) + \frac{31}{3 29} arctan (\frac{3}{29} (x - \frac{2}{3}))) : \frac{5}{6} ln 9 x^{2} - 12 x + 33 + \frac{10}{3 29} arctan (\frac{3 x - 2}{29}) + \frac{31}{29} arctan (\frac{3 x - 2}{29})

= \frac{5}{6} ln 9 x^{2} - 12 x + 33 + \frac{10}{3 29} arctan (\frac{3 x - 2}{29}) + \frac{31}{29} arctan (\frac{3 x - 2}{29})

Füge eine Konstante zur Lösung hinzu

= \frac{5}{6} ln 9 x^{2} - 12 x + 33 + \frac{10}{3 29} arctan (\frac{3 x - 2}{29}) + \frac{31}{29} arctan (\frac{3 x - 2}{29}) + C

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