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

\int_{0}^{1} 7 e^{- 4 x} d x

x \to 3 lim (lo g_{4} (3 x - 5))

x \to - 5 + lim (\frac{1}{x + 5})

\int \frac{9 x}{( 1 - x ^{2} ) ^{3}} d x

t an g e n t f (x) = 4 x^{3} 5 x - 1, at x = 1