integral of (5x^2-9)/(x^3-9x)

Lösung

\int \frac{5 x ^{2} - 9}{x ^{3} - 9 x} d x

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

Schritte zur Lösung

\int \frac{5 x ^{2} - 9}{x ^{3} - 9 x} d x

Multipliziere aus

\frac{5 x ^{2} - 9}{x ^{3} - 9 x} : \frac{5 x ^{2}}{x ^{3} - 9 x} - \frac{9}{x ^{3} - 9 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{5 x ^{2}}{x ^{3} - 9 x} d x - \int \frac{9}{x ^{3} - 9 x} d x

\int \frac{5 x ^{2}}{x ^{3} - 9 x} d x = \frac{5}{2} ln (∣ x + 3 ∣ ∣ x - 3 ∣)

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

d er i v a t i v e f (x) = \frac{x - 5}{x + 5}

d er i v a t i v e f (x) = \frac{( 1 - 5 x )}{1 + 5 x}

\frac{d}{d x} y + y = 6

\int (2 - 2 x)^{9} d x

y^{^{'}} - y = 11 t e^{2 t}, y (0) = 1