de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of 1/(x^8(1+x^2))

Lösung

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

Lösung

\frac{1}{x} - \frac{1}{3 x ^{3}} + \frac{1}{5 x ^{5}} - \frac{1}{7 x ^{7}} + arctan (x) + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{1}{x ^{8} ( 1 + x ^{2} )} : - \frac{1}{x ^{2}} + \frac{1}{x ^{4}} - \frac{1}{x ^{6}} + \frac{1}{x ^{8}} + \frac{1}{x ^{2} + 1}

= \int - \frac{1}{x ^{2}} + \frac{1}{x ^{4}} - \frac{1}{x ^{6}} + \frac{1}{x ^{8}} + \frac{1}{x ^{2} + 1} 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}{x ^{2}} d x + \int \frac{1}{x ^{4}} d x - \int \frac{1}{x ^{6}} d x + \int \frac{1}{x ^{8}} d x + \int \frac{1}{x ^{2} + 1} d x

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

\int \frac{1}{x ^{4}} d x = - \frac{1}{3 x ^{3}}

\int \frac{1}{x ^{6}} d x = - \frac{1}{5 x ^{5}}

\int \frac{1}{x ^{8}} d x = - \frac{1}{7 x ^{7}}

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

= - (- \frac{1}{x}) - \frac{1}{3 x ^{3}} - (- \frac{1}{5 x ^{5}}) - \frac{1}{7 x ^{7}} + arctan (x)

Vereinfache

= \frac{1}{x} - \frac{1}{3 x ^{3}} + \frac{1}{5 x ^{5}} - \frac{1}{7 x ^{7}} + arctan (x)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{x} - \frac{1}{3 x ^{3}} + \frac{1}{5 x ^{5}} - \frac{1}{7 x ^{7}} + arctan (x) + C

Graph

Beliebte Beispiele

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

\int \frac{2 x + 3}{x ^{3} + x - 2 x} d x

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

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

integral of 9e^{0.2x}

\int 9 e^{0.2 x} d x

integral of x^2cos(x^3-1)

\int x^{2} cos (x^{3} - 1) d x

integral sin(x^2)

in t e g r a l sin (x^{2})