integral of (5x-3)/((x+1)(x-3))

Lösung

\int \frac{5 x - 3}{( x + 1 ) ( x - 3 )} d x

2 ln ∣ x + 1 ∣ + 3 ln ∣ x - 3 ∣ + C

Schritte zur Lösung

\int \frac{5 x - 3}{( x + 1 ) ( x - 3 )} d x

Wende U-Substitution an

= \int \frac{5 u - 8}{u ( u - 4 )} d u

Ermittle den Partialbruch von

\frac{5 u - 8}{u ( u - 4 )} : \frac{2}{u} + \frac{3}{u - 4}

= \int \frac{2}{u} + \frac{3}{u - 4} d u

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}{u} d u + \int \frac{3}{u - 4} d u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

\int \frac{3}{u - 4} d u = 3 ln ∣ u - 4 ∣

= 2 ln ∣ u ∣ + 3 ln ∣ u - 4 ∣

Setze in

u = x + 1

ein

= 2 ln ∣ x + 1 ∣ + 3 ln ∣ x + 1 - 4 ∣

Vereinfache

= 2 ln ∣ x + 1 ∣ + 3 ln ∣ x - 3 ∣

Füge eine Konstante zur Lösung hinzu

= 2 ln ∣ x + 1 ∣ + 3 ln ∣ x - 3 ∣ + C

(y^{2} - 2 x y + 6 x) d x - (x^{2} - 2 x y + 2) d y = 0

\int_{1}^{e^{2}} \frac{ln ^{2} ( x ^{2} )}{x} d x

\int e^{9 x} cos (4 x) d x

\int \frac{1}{csc ( 2 x )} csc (2 x) cot (2 x) d x

\frac{d y}{d x} + 3 x^{2} y = sin (x) e^{- x^{3}}, y (0) = 1