de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of (x^2)/(x^4-16)

Lösung

\int \frac{x ^{2}}{x ^{4} - 16} d x

Lösung

\frac{1}{4} arctan (\frac{x}{2}) - \frac{1}{8} ln ∣ x + 2 ∣ + \frac{1}{8} ln ∣ x - 2 ∣ + C

Schritte zur Lösung

\int \frac{x ^{2}}{x ^{4} - 16} d x

Ermittle den Partialbruch von

\frac{x ^{2}}{x ^{4} - 16} : \frac{1}{2 ( x ^{2} + 4 )} - \frac{1}{8 ( x + 2 )} + \frac{1}{8 ( x - 2 )}

= \int \frac{1}{2 ( x ^{2} + 4 )} - \frac{1}{8 ( x + 2 )} + \frac{1}{8 ( x - 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}{2 ( x ^{2} + 4 )} d x - \int \frac{1}{8 ( x + 2 )} d x + \int \frac{1}{8 ( x - 2 )} d x

\int \frac{1}{2 ( x ^{2} + 4 )} d x = \frac{1}{4} arctan (\frac{x}{2})

\int \frac{1}{8 ( x + 2 )} d x = \frac{1}{8} ln ∣ x + 2 ∣

\int \frac{1}{8 ( x - 2 )} d x = \frac{1}{8} ln ∣ x - 2 ∣

= \frac{1}{4} arctan (\frac{x}{2}) - \frac{1}{8} ln ∣ x + 2 ∣ + \frac{1}{8} ln ∣ x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} arctan (\frac{x}{2}) - \frac{1}{8} ln ∣ x + 2 ∣ + \frac{1}{8} ln ∣ x - 2 ∣ + C

Graph

Beliebte Beispiele

integral of sec(x)(3sec(x)+7tan(x))

\int sec (x) (3 sec (x) + 7 tan (x)) d x

integral of x(x^2-4)^{7/2}

\int x (x^{2} - 4)^{\frac{7}{2}} d x

integral of (1-3x)^9

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

integral of (x+ln(1-x))/(x^2)

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

integral of (3/(x^2)+4-1/(4x^3))

\int (\frac{3}{x ^{2}} + 4 - \frac{1}{4 x ^{3}}) d x