integral of 1/((x^2-100)^2)

Lösung

\int \frac{1}{( x ^{2} - 100 ) ^{2}} d x

\frac{1}{4000} ln ∣ x + 10 ∣ - \frac{1}{400 ( x + 10 )} - \frac{1}{4000} ln ∣ x - 10 ∣ - \frac{1}{400 ( x - 10 )} + C

Schritte zur Lösung

\int \frac{1}{( x ^{2} - 100 ) ^{2}} d x

Ermittle den Partialbruch von

\frac{1}{( x ^{2} - 100 ) ^{2}} : \frac{1}{4000 ( x + 10 )} + \frac{1}{400 ( x + 10 ) ^{2}} - \frac{1}{4000 ( x - 10 )} + \frac{1}{400 ( x - 10 ) ^{2}}

= \int \frac{1}{4000 ( x + 10 )} + \frac{1}{400 ( x + 10 ) ^{2}} - \frac{1}{4000 ( x - 10 )} + \frac{1}{400 ( x - 10 ) ^{2}} d 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{1}{4000 ( x + 10 )} d x + \int \frac{1}{400 ( x + 10 ) ^{2}} d x - \int \frac{1}{4000 ( x - 10 )} d x + \int \frac{1}{400 ( x - 10 ) ^{2}} d x

\int \frac{1}{4000 ( x + 10 )} d x = \frac{1}{4000} ln ∣ x + 10 ∣

\int \frac{1}{400 ( x + 10 ) ^{2}} d x = - \frac{1}{400 ( x + 10 )}

\int \frac{1}{4000 ( x - 10 )} d x = \frac{1}{4000} ln ∣ x - 10 ∣

\int \frac{1}{400 ( x - 10 ) ^{2}} d x = - \frac{1}{400 ( x - 10 )}

= \frac{1}{4000} ln ∣ x + 10 ∣ - \frac{1}{400 ( x + 10 )} - \frac{1}{4000} ln ∣ x - 10 ∣ - \frac{1}{400 ( x - 10 )}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4000} ln ∣ x + 10 ∣ - \frac{1}{400 ( x + 10 )} - \frac{1}{4000} ln ∣ x - 10 ∣ - \frac{1}{400 ( x - 10 )} + C

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

a re a x = 0, y = 0, y = \frac{- x ^{2} + 8}{9}

θ \to 0 lim (\frac{θ}{sin ( θ )})

\frac{\partial}{\partial x} ((x - y) (1 - x y))

d er i v a t i v e 7 4 x