de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of (4x^2+x+17)/((x+1)(x^2+4))

Lösung

\int \frac{4 x ^{2} + x + 17}{( x + 1 ) ( x ^{2} + 4 )} d x

Lösung

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

Schritte zur Lösung

\int \frac{4 x ^{2} + x + 17}{( x + 1 ) ( x ^{2} + 4 )} d x

Ermittle den Partialbruch von

\frac{4 x ^{2} + x + 17}{( x + 1 ) ( x ^{2} + 4 )} : \frac{4}{x + 1} + \frac{1}{x ^{2} + 4}

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

\int \frac{4}{x + 1} d x = 4 ln ∣ x + 1 ∣

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (x-1)(sqrt(2-x))

\int (x - 1) (2 - x) d x

integral of ((x^2-2))/(x^3-6x+2)

\int \frac{( x ^{2} - 2 )}{x ^{3} - 6 x + 2} d x

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

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

integral of 1/(sqrt(8+4x-4x^2))

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

integral of e^{-7x}sin(2x)

\int e^{- 7 x} sin (2 x) d x