integral of (7x-23)/((x-2)(x-5))

Lösung

\int \frac{7 x - 23}{( x - 2 ) ( x - 5 )} d x

3 ln ∣ x - 2 ∣ + 4 ln ∣ x - 5 ∣ + C

Schritte zur Lösung

\int \frac{7 x - 23}{( x - 2 ) ( x - 5 )} d x

Wende U-Substitution an

= \int \frac{7 u - 9}{u ( u - 3 )} d u

Ermittle den Partialbruch von

\frac{7 u - 9}{u ( u - 3 )} : \frac{3}{u} + \frac{4}{u - 3}

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

\int \frac{3}{u} d u = 3 ln ∣ u ∣

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

= 3 ln ∣ u ∣ + 4 ln ∣ u - 3 ∣

Setze in

u = x - 2

ein

= 3 ln ∣ x - 2 ∣ + 4 ln ∣ x - 2 - 3 ∣

Vereinfache

= 3 ln ∣ x - 2 ∣ + 4 ln ∣ x - 5 ∣

Füge eine Konstante zur Lösung hinzu

= 3 ln ∣ x - 2 ∣ + 4 ln ∣ x - 5 ∣ + C

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

\int \frac{6 x ^{2} - 32 x + 41}{x - 4} d x

\int \frac{x ^{3}}{( x ^{2} - 4 ) ^{\frac{3}{2}}} d x

\int \frac{x}{( x ^{2} - 25 ) ^{3}} d x

\int lo g_{6} (x - 1) d x