de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of (23x^2)/((x+1)(x^2+1))

Lösung

\int \frac{23 x ^{2}}{( x + 1 ) ( x ^{2} + 1 )} d x

Lösung

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

Schritte zur Lösung

\int \frac{23 x ^{2}}{( x + 1 ) ( x ^{2} + 1 )} d x

Entferne die Konstante:

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

= 23 \cdot \int \frac{x ^{2}}{( x + 1 ) ( x ^{2} + 1 )} d x

Ermittle den Partialbruch von

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

= 23 \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

= 23 (\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))

= 23 (\frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} (\frac{1}{2} ln x^{2} + 1 - arctan (x)))

Füge eine Konstante zur Lösung hinzu

= 23 (\frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} (\frac{1}{2} ln x^{2} + 1 - arctan (x))) + C

Graph

Beliebte Beispiele

integral of (6x^2-3x)/((x-2)(x+4))

\int \frac{6 x ^{2} - 3 x}{( x - 2 ) ( x + 4 )} d x

integral of 2+4x+42x^2

\int 2 + 4 x + 42 x^{2} d x

integral of 2/((x+4)(x+6))

\int \frac{2}{( x + 4 ) ( x + 6 )} d x

integral of 3/(4x^2-1)

\int \frac{3}{4 x ^{2} - 1} d x

integral of 5/(xsqrt(4x^4-9))

\int \frac{5}{x 4 x ^{4} - 9} d x