de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Ermittle den Partialbruch von

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

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

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

\int \frac{1}{2 ( x - 1 )} d x = \frac{1}{2} ln ∣ x - 1 ∣

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

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

Vereinfache

2 (\frac{1}{2} ln ∣ x - 1 ∣ + \frac{1}{2} (- \frac{1}{2} ln x^{2} + 1 + arctan (x))) : ln ∣ x - 1 ∣ - \frac{1}{2} ln x^{2} + 1 + arctan (x)

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (1-x^2)^{-1}

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

integral of x^3ln(5x^2)

\int x^{3} ln (5 x^{2}) d x

integral of (5x-2)/(x(x-2))

\int \frac{5 x - 2}{x ( x - 2 )} d x

integral of 5ln(2x)

\int 5 ln (2 x) d x

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

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