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

Lösung

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

3 ln ∣ x + 2 ∣ + \frac{7}{x + 2} + C

Schritte zur Lösung

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

Multipliziere aus

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

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

\int \frac{3 x}{( x + 2 ) ^{2}} d x = 3 (ln ∣ x + 2 ∣ + \frac{2}{x + 2})

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

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

Vereinfache

3 (ln ∣ x + 2 ∣ + \frac{2}{x + 2}) - (- \frac{1}{x + 2}) : 3 ln ∣ x + 2 ∣ + \frac{7}{x + 2}

= 3 ln ∣ x + 2 ∣ + \frac{7}{x + 2}

Füge eine Konstante zur Lösung hinzu

= 3 ln ∣ x + 2 ∣ + \frac{7}{x + 2} + C

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