integral of (2x^2-25x-33)/((x+1)^2(x-5))

Lösung

\int \frac{2 x ^{2} - 25 x - 33}{( x + 1 ) ^{2} ( x - 5 )} d x

5 ln ∣ x + 1 ∣ + 2 ln ∣ x - 5 ∣ - \frac{29}{6} ln ∣ x - 5 ∣ - \frac{1}{x + 1} - \frac{1}{6} ln ∣ x - 5 ∣ + C

Schritte zur Lösung

\int \frac{2 x ^{2} - 25 x - 33}{( x + 1 ) ^{2} ( x - 5 )} d x

Wende U-Substitution an

= \int \frac{2 u ^{2} - 29 u - 6}{u ^{2} ( u - 6 )} d u

Multipliziere aus

\frac{2 u ^{2} - 29 u - 6}{u ^{2} ( u - 6 )} : \frac{2 u ^{2}}{u ^{2} ( u - 6 )} - \frac{29 u}{u ^{2} ( u - 6 )} - \frac{6}{u ^{2} ( u - 6 )}

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 ^{2}}{u ^{2} ( u - 6 )} d u - \int \frac{29 u}{u ^{2} ( u - 6 )} d u - \int \frac{6}{u ^{2} ( u - 6 )} d u

\int \frac{2 u ^{2}}{u ^{2} ( u - 6 )} d u = 2 ln ∣ u - 6 ∣

\int \frac{29 u}{u ^{2} ( u - 6 )} d u = 29 (- \frac{1}{6} ln ∣ u ∣ + \frac{1}{6} ln ∣ u - 6 ∣)

\int \frac{6}{u ^{2} ( u - 6 )} d u = - \frac{1}{6} ln ∣ u ∣ + \frac{1}{u} + \frac{1}{6} ln ∣ u - 6 ∣

= 2 ln ∣ u - 6 ∣ - 29 (- \frac{1}{6} ln ∣ u ∣ + \frac{1}{6} ln ∣ u - 6 ∣) - (- \frac{1}{6} ln ∣ u ∣ + \frac{1}{u} + \frac{1}{6} ln ∣ u - 6 ∣)

Setze in

u = x + 1

ein

= 2 ln ∣ x + 1 - 6 ∣ - 29 (- \frac{1}{6} ln ∣ x + 1 ∣ + \frac{1}{6} ln ∣ x + 1 - 6 ∣) - (- \frac{1}{6} ln ∣ x + 1 ∣ + \frac{1}{x + 1} + \frac{1}{6} ln ∣ x + 1 - 6 ∣)

Vereinfache

2 ln ∣ x + 1 - 6 ∣ - 29 (- \frac{1}{6} ln ∣ x + 1 ∣ + \frac{1}{6} ln ∣ x + 1 - 6 ∣) - (- \frac{1}{6} ln ∣ x + 1 ∣ + \frac{1}{x + 1} + \frac{1}{6} ln ∣ x + 1 - 6 ∣) : 5 ln ∣ x + 1 ∣ + 2 ln ∣ x - 5 ∣ - \frac{29}{6} ln ∣ x - 5 ∣ - \frac{1}{x + 1} - \frac{1}{6} ln ∣ x - 5 ∣

= 5 ln ∣ x + 1 ∣ + 2 ln ∣ x - 5 ∣ - \frac{29}{6} ln ∣ x - 5 ∣ - \frac{1}{x + 1} - \frac{1}{6} ln ∣ x - 5 ∣

Füge eine Konstante zur Lösung hinzu

= 5 ln ∣ x + 1 ∣ + 2 ln ∣ x - 5 ∣ - \frac{29}{6} ln ∣ x - 5 ∣ - \frac{1}{x + 1} - \frac{1}{6} ln ∣ x - 5 ∣ + C

3 e^{5 x} \frac{d y}{d x} = \frac{- 25 x}{y ^{2}}, y (0) = 1

x \to 2 lim ((x - 1) (x + 1))

a re a 2 x^{2} + 12, [- 5, 0]

\int_{1}^{2} (3 x^{- 2}) d x

\int 2 - 2 x^{2} d x