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

Lösung

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

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{x ^{2} + 6 x}{( x + 3 ) ^{2}} d x

\int \frac{0}{x + 2} d x

\int \frac{1}{5 + 4 cos ( 2 x )} d x

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

\int (e^{x} - 1 - x) d x