integral of (8x+10)/(2x^2+5x)

Lösung

\int \frac{8 x + 10}{2 x ^{2} + 5 x} d x

2 ln ∣ 2 x + 5 ∣ + 2 ln ∣ x ∣ + C

Schritte zur Lösung

\int \frac{8 x + 10}{2 x ^{2} + 5 x} d x

Multipliziere aus

\frac{8 x + 10}{2 x ^{2} + 5 x} : \frac{8 x}{2 x ^{2} + 5 x} + \frac{10}{2 x ^{2} + 5 x}

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{8 x}{2 x ^{2} + 5 x} d x + \int \frac{10}{2 x ^{2} + 5 x} d x

\int \frac{8 x}{2 x ^{2} + 5 x} d x = 4 ln ∣ 2 x + 5 ∣

\int \frac{10}{2 x ^{2} + 5 x} d x = 10 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{5} ln ∣ 2 x + 5 ∣)

= 4 ln ∣ 2 x + 5 ∣ + 10 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{5} ln ∣ 2 x + 5 ∣)

Vereinfache

4 ln ∣ 2 x + 5 ∣ + 10 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{5} ln ∣ 2 x + 5 ∣) : 2 ln ∣ 2 x + 5 ∣ + 2 ln ∣ x ∣

= 2 ln ∣ 2 x + 5 ∣ + 2 ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= 2 ln ∣ 2 x + 5 ∣ + 2 ln ∣ x ∣ + C

\int \frac{1}{x 256 x ^{2} - 256} d x

x \to 1 - lim ((x - 1)^{2} f (x))

d er i v a t i v e cot (x^{7})

x \to 2 lim (x 8 - x^{2})

\int \frac{1}{16 x ^{2} - 25} d x