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

Lösung

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

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

Schritte zur Lösung

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

Wende integrale Substitution an

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

Multipliziere aus

\frac{2 + 3 u}{u ^{2} + 1} : \frac{2}{u ^{2} + 1} + \frac{3 u}{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{2}{u ^{2} + 1} d u + \int \frac{3 u}{u ^{2} + 1} d u)

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

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

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

Setze in

u = \frac{x}{3}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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