integral of 1/((x^2-49)^2)

Lösung

\int \frac{1}{( x ^{2} - 49 ) ^{2}} d x

\frac{1}{1372} ln ∣ x + 7 ∣ - \frac{1}{196 ( x + 7 )} - \frac{1}{1372} ln ∣ x - 7 ∣ - \frac{1}{196 ( x - 7 )} + C

Schritte zur Lösung

\int \frac{1}{( x ^{2} - 49 ) ^{2}} d x

Ermittle den Partialbruch von

\frac{1}{( x ^{2} - 49 ) ^{2}} : \frac{1}{1372 ( x + 7 )} + \frac{1}{196 ( x + 7 ) ^{2}} - \frac{1}{1372 ( x - 7 )} + \frac{1}{196 ( x - 7 ) ^{2}}

= \int \frac{1}{1372 ( x + 7 )} + \frac{1}{196 ( x + 7 ) ^{2}} - \frac{1}{1372 ( x - 7 )} + \frac{1}{196 ( x - 7 ) ^{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}{1372 ( x + 7 )} d x + \int \frac{1}{196 ( x + 7 ) ^{2}} d x - \int \frac{1}{1372 ( x - 7 )} d x + \int \frac{1}{196 ( x - 7 ) ^{2}} d x

\int \frac{1}{1372 ( x + 7 )} d x = \frac{1}{1372} ln ∣ x + 7 ∣

\int \frac{1}{196 ( x + 7 ) ^{2}} d x = - \frac{1}{196 ( x + 7 )}

\int \frac{1}{1372 ( x - 7 )} d x = \frac{1}{1372} ln ∣ x - 7 ∣

\int \frac{1}{196 ( x - 7 ) ^{2}} d x = - \frac{1}{196 ( x - 7 )}

= \frac{1}{1372} ln ∣ x + 7 ∣ - \frac{1}{196 ( x + 7 )} - \frac{1}{1372} ln ∣ x - 7 ∣ - \frac{1}{196 ( x - 7 )}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{1372} ln ∣ x + 7 ∣ - \frac{1}{196 ( x + 7 )} - \frac{1}{1372} ln ∣ x - 7 ∣ - \frac{1}{196 ( x - 7 )} + C

d er i v a t i v e f (x) = 2 x + 2

x \to 4 lim (\frac{7}{x - 4})

x \to 0 lim (3 x^{2} + 5)

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

\int \frac{9 + x + x ^{2}}{x} d x