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Beliebt Rechnen >

integral of (8x+6)/(2x^2+3x)

Lösung

\int \frac{8 x + 6}{2 x ^{2} + 3 x} d x

Lösung

2 ln ∣ 2 x + 3 ∣ + 2 ln ∣ x ∣ + C

Schritte zur Lösung

\int \frac{8 x + 6}{2 x ^{2} + 3 x} d x

Multipliziere aus

\frac{8 x + 6}{2 x ^{2} + 3 x} : \frac{8 x}{2 x ^{2} + 3 x} + \frac{6}{2 x ^{2} + 3 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{8 x}{2 x ^{2} + 3 x} d x + \int \frac{6}{2 x ^{2} + 3 x} d x

\int \frac{8 x}{2 x ^{2} + 3 x} d x = 4 ln ∣ 2 x + 3 ∣

\int \frac{6}{2 x ^{2} + 3 x} d x = 6 (\frac{1}{3} ln ∣ x ∣ - \frac{1}{3} ln ∣ 2 x + 3 ∣)

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

Vereinfache

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

= 2 ln ∣ 2 x + 3 ∣ + 2 ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= 2 ln ∣ 2 x + 3 ∣ + 2 ln ∣ x ∣ + C

Graph

Beliebte Beispiele

integral of 1/(300+x)

\int \frac{1}{300 + x} d x

integral of (x^3)/(sqrt(x^8+9))

\int \frac{x ^{3}}{x ^{8} + 9} d x

integral of sin(3x+14)

\int sin (3 x + 14) d x

integral of x(x^2-3)^{2022}

\int x (x^{2} - 3)^{2022} d x

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

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