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Beliebt Rechnen >

integral of (2x-3)/(x^2+9)

Lösung

\int \frac{2 x - 3}{x ^{2} + 9} d x

Lösung

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

Schritte zur Lösung

\int \frac{2 x - 3}{x ^{2} + 9} d x

Wende integrale Substitution an

= \int \frac{2 u - 1}{u ^{2} + 1} d u

Multipliziere aus

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

= \int \frac{2 u}{u ^{2} + 1} d u - \int \frac{1}{u ^{2} + 1} d u

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

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

= ln u^{2} + 1 - arctan (u)

Setze in

u = \frac{x}{3}

ein

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

(\frac{x}{3})^{2} = \frac{x ^{2}}{9}

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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