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Beliebt Rechnen >

integral of (2x^2-3)/(x^2+5x)

Lösung

\int \frac{2 x ^{2} - 3}{x ^{2} + 5 x} d x

Lösung

2 x - \frac{3}{5} ln ∣ x ∣ - \frac{47}{5} ln ∣ x + 5 ∣ + 10 + C

Schritte zur Lösung

\int \frac{2 x ^{2} - 3}{x ^{2} + 5 x} d x

Multipliziere aus

\frac{2 x ^{2} - 3}{x ^{2} + 5 x} : \frac{2 x ^{2}}{x ^{2} + 5 x} - \frac{3}{x ^{2} + 5 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{2 x ^{2}}{x ^{2} + 5 x} d x - \int \frac{3}{x ^{2} + 5 x} d x

\int \frac{2 x ^{2}}{x ^{2} + 5 x} d x = 2 (x + 5 - 5 ln ∣ x + 5 ∣)

\int \frac{3}{x ^{2} + 5 x} d x = 3 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{5} ln ∣ x + 5 ∣)

= 2 (x + 5 - 5 ln ∣ x + 5 ∣) - 3 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{5} ln ∣ x + 5 ∣)

Vereinfache

2 (x + 5 - 5 ln ∣ x + 5 ∣) - 3 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{5} ln ∣ x + 5 ∣) : 2 x - \frac{3}{5} ln ∣ x ∣ - \frac{47}{5} ln ∣ x + 5 ∣ + 10

= 2 x - \frac{3}{5} ln ∣ x ∣ - \frac{47}{5} ln ∣ x + 5 ∣ + 10

Füge eine Konstante zur Lösung hinzu

= 2 x - \frac{3}{5} ln ∣ x ∣ - \frac{47}{5} ln ∣ x + 5 ∣ + 10 + C

Graph

Beliebte Beispiele

integral of (x^3)/(sqrt(4x^2+9))

\int \frac{x ^{3}}{4 x ^{2} + 9} d x

integral of (2x^3)/(e^{x^4)}

\int \frac{2 x ^{3}}{e ^{x^{4}}} d x

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

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

integral of 3^xcos(2x)

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

integral of (2x)/(x^2-5)

\int \frac{2 x}{x ^{2} - 5} d x