integral of (x+5)/(x^2+5x+6)

Lösung

\int \frac{x + 5}{x ^{2} + 5 x + 6} d x

2 (\frac{1}{4} ln 4 x^{2} + 20 x + 24 + \frac{5}{4} (ln ∣ 2 x + 6 ∣ - ln ∣ 2 x + 4 ∣)) + 5 (ln ∣ x + 2 ∣ - ln ∣ x + 3 ∣) + C

Schritte zur Lösung

\int \frac{x + 5}{x ^{2} + 5 x + 6} d x

Multipliziere aus

\frac{x + 5}{x ^{2} + 5 x + 6} : \frac{x}{x ^{2} + 5 x + 6} + \frac{5}{x ^{2} + 5 x + 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{x}{x ^{2} + 5 x + 6} d x + \int \frac{5}{x ^{2} + 5 x + 6} d x

\int \frac{x}{x ^{2} + 5 x + 6} d x = 2 (\frac{1}{4} ln 4 x^{2} + 20 x + 24 + \frac{5}{4} (ln ∣ 2 x + 6 ∣ - ln ∣ 2 x + 4 ∣))

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

= 2 (\frac{1}{4} ln 4 x^{2} + 20 x + 24 + \frac{5}{4} (ln ∣ 2 x + 6 ∣ - ln ∣ 2 x + 4 ∣)) + 5 (ln ∣ x + 2 ∣ - ln ∣ x + 3 ∣)

Füge eine Konstante zur Lösung hinzu

= 2 (\frac{1}{4} ln 4 x^{2} + 20 x + 24 + \frac{5}{4} (ln ∣ 2 x + 6 ∣ - ln ∣ 2 x + 4 ∣)) + 5 (ln ∣ x + 2 ∣ - ln ∣ x + 3 ∣) + C

a re a y = x^{2} - 9, y = 8 x

l a pl a ce t r an s f or m e^{2 t} sin (j) (2 t)

\int e^{9 x + 1} d x

\frac{\partial}{\partial x} (- 3 x^{3} y^{2} sin (x^{3} y^{3}))

x y^{^{'}} + y = e^{x}, y (1) = 7