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

Lösung

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

9 x + 13 ln ∣ x - 3 ∣ + \frac{1}{2} (x - 3)^{2} - 27 + C

Schritte zur Lösung

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

Multipliziere aus

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

\int \frac{5}{x - 3} d x = 5 ln ∣ x - 3 ∣

= \frac{( x - 3 ) ^{2}}{2} + 6 (x - 3) + 9 ln ∣ x - 3 ∣ + 3 (x - 3 + 3 ln ∣ x - 3 ∣) - 5 ln ∣ x - 3 ∣

Vereinfache

\frac{( x - 3 ) ^{2}}{2} + 6 (x - 3) + 9 ln ∣ x - 3 ∣ + 3 (x - 3 + 3 ln ∣ x - 3 ∣) - 5 ln ∣ x - 3 ∣ : 9 x + 13 ln ∣ x - 3 ∣ + \frac{1}{2} (x - 3)^{2} - 27

= 9 x + 13 ln ∣ x - 3 ∣ + \frac{1}{2} (x - 3)^{2} - 27

Füge eine Konstante zur Lösung hinzu

= 9 x + 13 ln ∣ x - 3 ∣ + \frac{1}{2} (x - 3)^{2} - 27 + C

\int 2 + x^{2} d x

\frac{d}{d x} (\frac{y}{3 x})

\frac{d y}{d x} = - 2 y, y (0) = 3

\frac{d}{d x} ((3 x^{2} + ln (x)) \cdot (5 x^{3})^{x})

\int 7 sin^{5} (x) cos (x) d x