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

Lösung

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

\frac{x ^{2}}{2} - x + \frac{2}{3} ln ∣ 3 x + 2 ∣ - \frac{2}{9} + C

Schritte zur Lösung

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

Multipliziere aus

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

\int \frac{3 x ^{2}}{3 x + 2} d x = 3 (\frac{x ^{2}}{6} - \frac{2}{9} x + \frac{4}{27} ln ∣ 3 x + 2 ∣)

\int \frac{x}{3 x + 2} d x = \frac{1}{9} (3 x + 2 - 2 ln ∣ 3 x + 2 ∣)

= 3 (\frac{x ^{2}}{6} - \frac{2}{9} x + \frac{4}{27} ln ∣ 3 x + 2 ∣) - \frac{1}{9} (3 x + 2 - 2 ln ∣ 3 x + 2 ∣)

Vereinfache

3 (\frac{x ^{2}}{6} - \frac{2}{9} x + \frac{4}{27} ln ∣ 3 x + 2 ∣) - \frac{1}{9} (3 x + 2 - 2 ln ∣ 3 x + 2 ∣) : \frac{x ^{2}}{2} - x + \frac{2}{3} ln ∣ 3 x + 2 ∣ - \frac{2}{9}

= \frac{x ^{2}}{2} - x + \frac{2}{3} ln ∣ 3 x + 2 ∣ - \frac{2}{9}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{2}}{2} - x + \frac{2}{3} ln ∣ 3 x + 2 ∣ - \frac{2}{9} + C

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

\int sin^{3} (2 t) d t

\int x^{2} + x y d y

\int (tan (a x) - cot (a x))^{3} d x

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