integral of 3/(x(x^2+1))

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Ermittle den Partialbruch von

\frac{1}{x ( x ^{2} + 1 )} : \frac{1}{x} - \frac{x}{x ^{2} + 1}

= 3 \cdot \int \frac{1}{x} - \frac{x}{x ^{2} + 1} d x

Wende die Summenregel an:

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

Füge eine Konstante zur Lösung hinzu

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

\frac{d}{d x} (∣ (- 3)^{x} ∣)

y^{^{'}} x^{2} + y x^{2} = x y^{5}

d er i v a t i v e f (x) = \frac{1}{2}

d er i v a t i v e \frac{2 x ^{2} - 1}{x + 1}

d er i v a t i v e t^{3}