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^3+x^2+2)/((x^2+2)^2)

Lösung

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

Lösung

\frac{1}{2} ln x^{2} + 2 + \frac{1}{2} arctan (\frac{x}{2}) + \frac{1}{x ^{2} + 2} + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x ^{3} + x ^{2} + 2}{( x ^{2} + 2 ) ^{2}} : \frac{x + 1}{x ^{2} + 2} - \frac{2 x}{( x ^{2} + 2 ) ^{2}}

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

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

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

= \frac{1}{2} ln x^{2} + 2 + \frac{1}{2} arctan (\frac{x}{2}) - (- \frac{1}{x ^{2} + 2})

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln x^{2} + 2 + \frac{1}{2} arctan (\frac{x}{2}) + \frac{1}{x ^{2} + 2} + C

Graph

Beliebte Beispiele

integral of 1/(-e^x)

\int \frac{1}{- e ^{x}} d x

integral of 3/(1-cos(x))

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

integral of ln(2t+1)

\int ln (2 t + 1) d t

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

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

integral of 3/(x^2+14x+97)

\int \frac{3}{x ^{2} + 14 x + 97} d x