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^4+9x^2+x+2)/(x^2+9)

Lösung

\int \frac{x ^{4} + 9 x ^{2} + x + 2}{x ^{2} + 9} d x

Lösung

\frac{x ^{3}}{3} + \frac{1}{2} ln x^{2} + 9 + \frac{2}{3} arctan (\frac{x}{3}) + C

Schritte zur Lösung

\int \frac{x ^{4} + 9 x ^{2} + x + 2}{x ^{2} + 9} d x

Multipliziere aus

\frac{x ^{4} + 9 x ^{2} + x + 2}{x ^{2} + 9} : \frac{x ^{4}}{x ^{2} + 9} + \frac{9 x ^{2}}{x ^{2} + 9} + \frac{x}{x ^{2} + 9} + \frac{2}{x ^{2} + 9}

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} + 9} d x + \int \frac{9 x ^{2}}{x ^{2} + 9} d x + \int \frac{x}{x ^{2} + 9} d x + \int \frac{2}{x ^{2} + 9} d x

\int \frac{x ^{4}}{x ^{2} + 9} d x = 27 (\frac{x ^{3}}{81} - \frac{x}{3} + arctan (\frac{x}{3}))

\int \frac{9 x ^{2}}{x ^{2} + 9} d x = 27 (- arctan (\frac{x}{3}) + \frac{x}{3})

\int \frac{x}{x ^{2} + 9} d x = \frac{1}{2} ln x^{2} + 9

\int \frac{2}{x ^{2} + 9} d x = \frac{2}{3} arctan (\frac{x}{3})

= 27 (\frac{x ^{3}}{81} - \frac{x}{3} + arctan (\frac{x}{3})) + 27 (- arctan (\frac{x}{3}) + \frac{x}{3}) + \frac{1}{2} ln x^{2} + 9 + \frac{2}{3} arctan (\frac{x}{3})

Vereinfache

27 (\frac{x ^{3}}{81} - \frac{x}{3} + arctan (\frac{x}{3})) + 27 (- arctan (\frac{x}{3}) + \frac{x}{3}) + \frac{1}{2} ln x^{2} + 9 + \frac{2}{3} arctan (\frac{x}{3}) : \frac{x ^{3}}{3} + \frac{1}{2} ln x^{2} + 9 + \frac{2}{3} arctan (\frac{x}{3})

= \frac{x ^{3}}{3} + \frac{1}{2} ln x^{2} + 9 + \frac{2}{3} arctan (\frac{x}{3})

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{3}}{3} + \frac{1}{2} ln x^{2} + 9 + \frac{2}{3} arctan (\frac{x}{3}) + C

Graph

Beliebte Beispiele

limit as x approaches-2 of (2-x)/(2+x)

x \to - 2 lim (\frac{2 - x}{2 + x})

normal y=(x-2)^{2/3},(3,1)

n or ma l y = (x - 2)^{\frac{2}{3}}, (3, 1)

integral of e^{-1/3 x}

\int e^{- \frac{1}{3} x} d x

derivative of cot(e^x)

\frac{d}{d x} (cot (e^{x}))

sum from n=3 to infinity of (4/10)^n

n = 3 \sum \infty (\frac{4}{10})^{n}