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

Lösung

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

ln ∣ x ∣ - ln ∣ 3 x + 1 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{1}{x ( 1 + 3 x )} : \frac{1}{x} - \frac{3}{3 x + 1}

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

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{3}{3 x + 1} d x = ln ∣ 3 x + 1 ∣

= ln ∣ x ∣ - ln ∣ 3 x + 1 ∣

Füge eine Konstante zur Lösung hinzu

= ln ∣ x ∣ - ln ∣ 3 x + 1 ∣ + C

\int \frac{1}{x ( 1 + x ^{8} )} d x

\int \frac{1}{( 4 - x )} d x

\int p (p + 8)^{7} d p

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

\int (10 x + 1) 4 x - 1 d x