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

Lösung

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

20 (3 (\frac{1}{200} ln 100 x^{2} - 60 x + 40 + \frac{3}{100 31} arctan (\frac{10 x - 3}{31})) - \frac{1}{5 31} arctan (\frac{10 x - 3}{31})) + C

Schritte zur Lösung

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

Vervollständige das Quadrat

5 x^{2} - 3 x + 2 : 5 (x - \frac{3}{10})^{2} + \frac{31}{20}

= \int \frac{3 x - 2}{5 ( x - \frac{3}{10} ) ^{2} + \frac{31}{20}} d x

Wende U-Substitution an

= \int \frac{20 ( 3 ( u + \frac{3}{10} ) - 2 )}{100 u ^{2} + 31} d u

Entferne die Konstante:

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

= 20 \cdot \int \frac{3 ( u + \frac{3}{10} ) - 2}{100 u ^{2} + 31} d u

Multipliziere aus

\frac{3 ( u + \frac{3}{10} ) - 2}{100 u ^{2} + 31} : \frac{3 ( u + \frac{3}{10} )}{100 u ^{2} + 31} - \frac{2}{100 u ^{2} + 31}

Wende die Summenregel an:

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

= 20 (\int \frac{3 ( u + \frac{3}{10} )}{100 u ^{2} + 31} d u - \int \frac{2}{100 u ^{2} + 31} d u)

\int \frac{3 ( u + \frac{3}{10} )}{100 u ^{2} + 31} d u = 3 (\frac{1}{200} ln 100 u^{2} + 31 + \frac{3}{100 31} arctan (\frac{10}{31} u))

\int \frac{2}{100 u ^{2} + 31} d u = \frac{1}{5 31} arctan (\frac{10}{31} u)

= 20 (3 (\frac{1}{200} ln 100 u^{2} + 31 + \frac{3}{100 31} arctan (\frac{10}{31} u)) - \frac{1}{5 31} arctan (\frac{10}{31} u))

Setze in

u = x - \frac{3}{10}

ein

= 20 (3 (\frac{1}{200} ln 100 (x - \frac{3}{10})^{2} + 31 + \frac{3}{100 31} arctan (\frac{10}{31} (x - \frac{3}{10}))) - \frac{1}{5 31} arctan (\frac{10}{31} (x - \frac{3}{10})))

Vereinfache

20 (3 (\frac{1}{200} ln 100 (x - \frac{3}{10})^{2} + 31 + \frac{3}{100 31} arctan (\frac{10}{31} (x - \frac{3}{10}))) - \frac{1}{5 31} arctan (\frac{10}{31} (x - \frac{3}{10}))) : 20 (3 (\frac{1}{200} ln 100 x^{2} - 60 x + 40 + \frac{3}{100 31} arctan (\frac{10 x - 3}{31})) - \frac{1}{5 31} arctan (\frac{10 x - 3}{31}))

= 20 (3 (\frac{1}{200} ln 100 x^{2} - 60 x + 40 + \frac{3}{100 31} arctan (\frac{10 x - 3}{31})) - \frac{1}{5 31} arctan (\frac{10 x - 3}{31}))

Füge eine Konstante zur Lösung hinzu

= 20 (3 (\frac{1}{200} ln 100 x^{2} - 60 x + 40 + \frac{3}{100 31} arctan (\frac{10 x - 3}{31})) - \frac{1}{5 31} arctan (\frac{10 x - 3}{31})) + C

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