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^2+4))/((x-5)(x^2+9))

Lösung

\int \frac{( x ^{2} + 4 )}{( x - 5 ) ( x ^{2} + 9 )} d x

Lösung

\frac{9}{68} ln x^{2} + 9 + \frac{29}{34} ln ∣ x - 5 ∣ + \frac{15}{34} arctan (\frac{x}{3}) + \frac{1}{51} (- 3 ln x^{2} + 9 - 10 arctan (\frac{x}{3})) + C

Schritte zur Lösung

\int \frac{x ^{2} + 4}{( x - 5 ) ( x ^{2} + 9 )} d x

Multipliziere aus

\frac{x ^{2} + 4}{( x - 5 ) ( x ^{2} + 9 )} : \frac{x ^{2}}{( x - 5 ) ( x ^{2} + 9 )} + \frac{4}{( x - 5 ) ( 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 ^{2}}{( x - 5 ) ( x ^{2} + 9 )} d x + \int \frac{4}{( x - 5 ) ( x ^{2} + 9 )} d x

\int \frac{x ^{2}}{( x - 5 ) ( x ^{2} + 9 )} d x = \frac{25}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (\frac{9}{2} ln x^{2} + 9 + 15 arctan (\frac{x}{3}))

\int \frac{4}{( x - 5 ) ( x ^{2} + 9 )} d x = 4 (\frac{1}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (- \frac{1}{2} ln x^{2} + 9 - \frac{5}{3} arctan (\frac{x}{3})))

= \frac{25}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (\frac{9}{2} ln x^{2} + 9 + 15 arctan (\frac{x}{3})) + 4 (\frac{1}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (- \frac{1}{2} ln x^{2} + 9 - \frac{5}{3} arctan (\frac{x}{3})))

Vereinfache

\frac{25}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (\frac{9}{2} ln x^{2} + 9 + 15 arctan (\frac{x}{3})) + 4 (\frac{1}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (- \frac{1}{2} ln x^{2} + 9 - \frac{5}{3} arctan (\frac{x}{3}))) : \frac{9}{68} ln x^{2} + 9 + \frac{29}{34} ln ∣ x - 5 ∣ + \frac{15}{34} arctan (\frac{x}{3}) + \frac{1}{51} (- 3 ln x^{2} + 9 - 10 arctan (\frac{x}{3}))

= \frac{9}{68} ln x^{2} + 9 + \frac{29}{34} ln ∣ x - 5 ∣ + \frac{15}{34} arctan (\frac{x}{3}) + \frac{1}{51} (- 3 ln x^{2} + 9 - 10 arctan (\frac{x}{3}))

Füge eine Konstante zur Lösung hinzu

= \frac{9}{68} ln x^{2} + 9 + \frac{29}{34} ln ∣ x - 5 ∣ + \frac{15}{34} arctan (\frac{x}{3}) + \frac{1}{51} (- 3 ln x^{2} + 9 - 10 arctan (\frac{x}{3})) + C

Graph

Beliebte Beispiele

integral of x^4e^{(1-x^5)}

\int x^{4} e^{(1 - x^{5})} d x

\frac{d y}{d x} + 5 y = 8

derivative of sin(x^2+sin(x^2+sin(x^2)))

\frac{d}{d x} (sin (x^{2} + sin (x^{2} + sin (x^{2}))))

integral of 16x^2

\int 16 x^{2} d x

derivative of x|x|^{|x|-1}(ln|x|+1)

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