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Beliebt Rechnen >

integral of 1/((4-x^2)^2)

Lösung

\int \frac{1}{( 4 - x ^{2} ) ^{2}} d x

Lösung

\frac{1}{32} ln ∣ x + 2 ∣ - \frac{1}{16 ( x + 2 )} - \frac{1}{32} ln ∣ x - 2 ∣ - \frac{1}{16 ( x - 2 )} + C

Schritte zur Lösung

\int \frac{1}{( 4 - x ^{2} ) ^{2}} d x

Ermittle den Partialbruch von

\frac{1}{( 4 - x ^{2} ) ^{2}} : \frac{1}{32 ( x + 2 )} + \frac{1}{16 ( x + 2 ) ^{2}} - \frac{1}{32 ( x - 2 )} + \frac{1}{16 ( x - 2 ) ^{2}}

= \int \frac{1}{32 ( x + 2 )} + \frac{1}{16 ( x + 2 ) ^{2}} - \frac{1}{32 ( x - 2 )} + \frac{1}{16 ( x - 2 ) ^{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}{32 ( x + 2 )} d x + \int \frac{1}{16 ( x + 2 ) ^{2}} d x - \int \frac{1}{32 ( x - 2 )} d x + \int \frac{1}{16 ( x - 2 ) ^{2}} d x

\int \frac{1}{32 ( x + 2 )} d x = \frac{1}{32} ln ∣ x + 2 ∣

\int \frac{1}{16 ( x + 2 ) ^{2}} d x = - \frac{1}{16 ( x + 2 )}

\int \frac{1}{32 ( x - 2 )} d x = \frac{1}{32} ln ∣ x - 2 ∣

\int \frac{1}{16 ( x - 2 ) ^{2}} d x = - \frac{1}{16 ( x - 2 )}

= \frac{1}{32} ln ∣ x + 2 ∣ - \frac{1}{16 ( x + 2 )} - \frac{1}{32} ln ∣ x - 2 ∣ - \frac{1}{16 ( x - 2 )}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{32} ln ∣ x + 2 ∣ - \frac{1}{16 ( x + 2 )} - \frac{1}{32} ln ∣ x - 2 ∣ - \frac{1}{16 ( x - 2 )} + C

Graph

Beliebte Beispiele

(\partial)/(\partial x)(5ln(1+x^2-y))

\frac{\partial}{\partial x} (5 ln (1 + x^{2} - y))

limit as x approaches 0 of (ln(x))/(x^2)

x \to 0 lim (\frac{ln ( x )}{x ^{2}})

integral of 1/(sqrt(1+x+x^2))

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

f^'(x)=5(1+e^x)^4e^x

f^{^{'}} (x) = 5 (1 + e^{x})^{4} e^{x}

derivative h(w)=(5w^3-w)/w

d er i v a t i v e h (w) = \frac{5 w ^{3} - w}{w}