integral of (x^2-7x+12)/(x-3)

Lösung

\int \frac{x ^{2} - 7 x + 12}{x - 3} d x

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

Schritte zur Lösung

\int \frac{x ^{2} - 7 x + 12}{x - 3} d x

Multipliziere aus

\frac{x ^{2} - 7 x + 12}{x - 3} : \frac{x ^{2}}{x - 3} - \frac{7 x}{x - 3} + \frac{12}{x - 3}

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{x ^{2}}{x - 3} d x - \int \frac{7 x}{x - 3} d x + \int \frac{12}{x - 3} d x

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

\int \frac{7 x}{x - 3} d x = 7 (x - 3 + 3 ln ∣ x - 3 ∣)

\int \frac{12}{x - 3} d x = 12 ln ∣ x - 3 ∣

= \frac{( x - 3 ) ^{2}}{2} + 6 (x - 3) + 9 ln ∣ x - 3 ∣ - 7 (x - 3 + 3 ln ∣ x - 3 ∣) + 12 ln ∣ x - 3 ∣

Vereinfache

\frac{( x - 3 ) ^{2}}{2} + 6 (x - 3) + 9 ln ∣ x - 3 ∣ - 7 (x - 3 + 3 ln ∣ x - 3 ∣) + 12 ln ∣ x - 3 ∣ : - x + \frac{1}{2} (x - 3)^{2} + 3

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

Füge eine Konstante zur Lösung hinzu

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

d er i v a t i v e y = \frac{3 x - 9}{2 x + 8}

\frac{d}{d x} (\frac{4 + ln ( 2 x )}{2020 + x})

t a y l or e^{4 x}, 3

d er i v a t i v e 7 arctan (x - 1 + x^{2})

l a pl a ce t r an s f or m 8 e^{- 6 t} cos (4 t + 30)