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

Lösung

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

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{3 x + 1}{x ( x + 1 ) ^{2}} : \frac{1}{x} - \frac{1}{x + 1} + \frac{2}{( x + 1 ) ^{2}}

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

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

Füge eine Konstante zur Lösung hinzu

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

l a pl a ce t r an s f or m e^{- 2 t - 5}

\frac{\partial}{\partial x} (y \cdot ln (x))

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

\int_{0}^{π} sin^{2} (x) d x

\frac{\partial}{\partial x} (ln (4 - x^{2}))