integral of ((2x-7))/((x^{(2))+9)}

Lösung

\int \frac{( 2 x - 7 )}{( x ^{(2)} + 9 )} d x

ln \frac{x ^{2}}{9} + 1 - \frac{7}{3} arctan (\frac{x}{3}) + C

Schritte zur Lösung

\int \frac{2 x - 7}{x ^{2} + 9} d x

Wende integrale Substitution an

= \int \frac{6 u - 7}{3 ( u ^{2} + 1 )} d u

Entferne die Konstante:

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

= \frac{1}{3} \cdot \int \frac{6 u - 7}{u ^{2} + 1} d u

Multipliziere aus

\frac{6 u - 7}{u ^{2} + 1} : \frac{6 u}{u ^{2} + 1} - \frac{7}{u ^{2} + 1}

Wende die Summenregel an:

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

= \frac{1}{3} (\int \frac{6 u}{u ^{2} + 1} d u - \int \frac{7}{u ^{2} + 1} d u)

\int \frac{6 u}{u ^{2} + 1} d u = 3 ln u^{2} + 1

\int \frac{7}{u ^{2} + 1} d u = 7 arctan (u)

= \frac{1}{3} (3 ln u^{2} + 1 - 7 arctan (u))

Setze in

u = \frac{x}{3}

ein

= \frac{1}{3} (3 ln (\frac{x}{3})^{2} + 1 - 7 arctan (\frac{x}{3}))

Vereinfache

\frac{1}{3} (3 ln (\frac{x}{3})^{2} + 1 - 7 arctan (\frac{x}{3})) : ln \frac{x ^{2}}{9} + 1 - \frac{7}{3} arctan (\frac{x}{3})

= ln \frac{x ^{2}}{9} + 1 - \frac{7}{3} arctan (\frac{x}{3})

Füge eine Konstante zur Lösung hinzu

= ln \frac{x ^{2}}{9} + 1 - \frac{7}{3} arctan (\frac{x}{3}) + C

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

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