integral of 1/((x^2-36)^2)

Lösung

\int \frac{1}{( x ^{2} - 36 ) ^{2}} d x

\frac{1}{864} ln ∣ x + 6 ∣ - \frac{1}{144 ( x + 6 )} - \frac{1}{864} ln ∣ x - 6 ∣ - \frac{1}{144 ( x - 6 )} + C

Schritte zur Lösung

\int \frac{1}{( x ^{2} - 36 ) ^{2}} d x

Ermittle den Partialbruch von

\frac{1}{( x ^{2} - 36 ) ^{2}} : \frac{1}{864 ( x + 6 )} + \frac{1}{144 ( x + 6 ) ^{2}} - \frac{1}{864 ( x - 6 )} + \frac{1}{144 ( x - 6 ) ^{2}}

= \int \frac{1}{864 ( x + 6 )} + \frac{1}{144 ( x + 6 ) ^{2}} - \frac{1}{864 ( x - 6 )} + \frac{1}{144 ( x - 6 ) ^{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}{864 ( x + 6 )} d x + \int \frac{1}{144 ( x + 6 ) ^{2}} d x - \int \frac{1}{864 ( x - 6 )} d x + \int \frac{1}{144 ( x - 6 ) ^{2}} d x

\int \frac{1}{864 ( x + 6 )} d x = \frac{1}{864} ln ∣ x + 6 ∣

\int \frac{1}{144 ( x + 6 ) ^{2}} d x = - \frac{1}{144 ( x + 6 )}

\int \frac{1}{864 ( x - 6 )} d x = \frac{1}{864} ln ∣ x - 6 ∣

\int \frac{1}{144 ( x - 6 ) ^{2}} d x = - \frac{1}{144 ( x - 6 )}

= \frac{1}{864} ln ∣ x + 6 ∣ - \frac{1}{144 ( x + 6 )} - \frac{1}{864} ln ∣ x - 6 ∣ - \frac{1}{144 ( x - 6 )}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{864} ln ∣ x + 6 ∣ - \frac{1}{144 ( x + 6 )} - \frac{1}{864} ln ∣ x - 6 ∣ - \frac{1}{144 ( x - 6 )} + C

\int \frac{( e ^{2 x} )}{25 + e ^{4 x}} d x

\int - \frac{1}{262144} cos (8 x) d x

\int cos^{3} (x) cos (3 x) d x

n \to \infty lim (\frac{1}{n ^{2} + 2 n - n})

d er i v a t i v e \frac{sec ( x )}{7 + sec ( x )}