integral of (x+1)/(x^2+9)

Lösung

\int \frac{x + 1}{x ^{2} + 9} d x

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

Schritte zur Lösung

\int \frac{x + 1}{x ^{2} + 9} d x

Multipliziere aus

\frac{x + 1}{x ^{2} + 9} : \frac{x}{x ^{2} + 9} + \frac{1}{x ^{2} + 9}

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{x}{x ^{2} + 9} d x + \int \frac{1}{x ^{2} + 9} d x

\int \frac{x}{x ^{2} + 9} d x = \frac{1}{2} ln x^{2} + 9

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

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

Füge eine Konstante zur Lösung hinzu

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

\frac{d}{d x} (cot (x) tan (x))

x \to 0 + lim (∣ x ∣)

d er i v a t i v e f (t) = - 3 t^{2} + 2 t - 4

\frac{d}{d x} (ln (1 + x^{2}) - arcsin (\frac{1}{x}))

\int (12 x^{11} - \frac{11}{x ^{11}}) d x