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+4x^2-4x-1)/((x^2+1)^2)

Lösung

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

Lösung

\frac{1}{2} ln x^{2} + 1 + \frac{3}{2} arctan (x) - \frac{5}{2} \cdot \frac{x}{x ^{2} + 1} + \frac{5}{2 ( x ^{2} + 1 )} + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x ^{3} + 4 x ^{2} - 4 x - 1}{( x ^{2} + 1 ) ^{2}} : \frac{x + 4}{x ^{2} + 1} + \frac{- 5 x - 5}{( x ^{2} + 1 ) ^{2}}

= \int \frac{x + 4}{x ^{2} + 1} + \frac{- 5 x - 5}{( x ^{2} + 1 ) ^{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 + 4}{x ^{2} + 1} d x + \int \frac{- 5 x - 5}{( x ^{2} + 1 ) ^{2}} d x

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

\int \frac{- 5 x - 5}{( x ^{2} + 1 ) ^{2}} d x = \frac{5}{2 ( x ^{2} + 1 )} - \frac{5}{2} (arctan (x) + \frac{x}{1 + x ^{2}})

= \frac{1}{2} ln x^{2} + 1 + 4 arctan (x) + \frac{5}{2 ( x ^{2} + 1 )} - \frac{5}{2} (arctan (x) + \frac{x}{1 + x ^{2}})

Vereinfache

\frac{1}{2} ln x^{2} + 1 + 4 arctan (x) + \frac{5}{2 ( x ^{2} + 1 )} - \frac{5}{2} (arctan (x) + \frac{x}{1 + x ^{2}}) : \frac{1}{2} ln x^{2} + 1 + \frac{3}{2} arctan (x) - \frac{5}{2} \cdot \frac{x}{x ^{2} + 1} + \frac{5}{2 ( x ^{2} + 1 )}

= \frac{1}{2} ln x^{2} + 1 + \frac{3}{2} arctan (x) - \frac{5}{2} \cdot \frac{x}{x ^{2} + 1} + \frac{5}{2 ( x ^{2} + 1 )}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln x^{2} + 1 + \frac{3}{2} arctan (x) - \frac{5}{2} \cdot \frac{x}{x ^{2} + 1} + \frac{5}{2 ( x ^{2} + 1 )} + C

Graph

Beliebte Beispiele

integral of (x^2+31)/(x^2+36)

\int \frac{x ^{2} + 31}{x ^{2} + 36} d x

integral of 1/(x^2-6x-7)

\int \frac{1}{x ^{2} - 6 x - 7} d x

integral 1/(sec^2(x))

in t e g r a l \frac{1}{sec ^{2} ( x )}

integral of 5x^2-2x+4

\int 5 x^{2} - 2 x + 4 d x

integral of (ln(x))/x (1+ln(x))

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