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

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

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

= - \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}

= - \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

= - (\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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

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

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\int \frac{3}{x + 5} d x

\int (\frac{4}{x ^{5}} - \frac{5}{x ^{2}} + \frac{7}{x}) d x

\int (e^{7 x} + 8) (e^{- 4 x} + 21) d x