de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of (25)/((x-1)(x^2+4))

Lösung

\int \frac{25}{( x - 1 ) ( x ^{2} + 4 )} d x

Lösung

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

Schritte zur Lösung

\int \frac{25}{( x - 1 ) ( x ^{2} + 4 )} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 25 \cdot \int \frac{1}{( x - 1 ) ( x ^{2} + 4 )} d x

Ermittle den Partialbruch von

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

= 25 \cdot \int \frac{1}{5 ( x - 1 )} + \frac{- x - 1}{5 ( 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

= 25 (\int \frac{1}{5 ( x - 1 )} d x + \int \frac{- x - 1}{5 ( x ^{2} + 4 )} d x)

\int \frac{1}{5 ( x - 1 )} d x = \frac{1}{5} ln ∣ x - 1 ∣

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

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

Füge eine Konstante zur Lösung hinzu

= 25 (\frac{1}{5} ln ∣ x - 1 ∣ + \frac{1}{5} (- \frac{1}{2} ln x^{2} + 4 - \frac{1}{2} arctan (\frac{x}{2}))) + C

Graph

Beliebte Beispiele

integral of (x^2+1)^2x

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

derivative ln(1-3x)

d er i v a t i v e ln (1 - 3 x)

integral of x^2n^2xx

\int x^{2} n^{2} xx d x

integral of x(3-x)

\int x (3 - x) d x

integral of (5e^x)/(e^{2x)+6e^x+9}

\int \frac{5 e ^{x}}{e ^{2 x} + 6 e ^{x} + 9} d x