integral of (-x^2-1)/(x^2-1)

Lösung

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

- x + ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣ + C

Schritte zur Lösung

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

Multipliziere aus

\frac{- x ^{2} - 1}{x ^{2} - 1} : - \frac{x ^{2}}{x ^{2} - 1} - \frac{1}{x ^{2} - 1}

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

\int \frac{x ^{2}}{x ^{2} - 1} d x = - \frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} ln ∣ x - 1 ∣ + x

\int \frac{1}{x ^{2} - 1} d x = - \frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} ln ∣ x - 1 ∣

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

Vereinfache

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

= - x + ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣

Füge eine Konstante zur Lösung hinzu

= - x + ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣ + C

d er i v a t i v e 9 e^{x} (cos (x) - sin (x))

n = 0 \sum \infty (- x)^{n}

x \to 0 + lim (\frac{x}{ln ( \frac{1}{x} )})

\frac{\partial}{\partial x} (\frac{y}{2 x})

d er i v a t i v e x \cdot sin (\frac{4}{x})