de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

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

Lösung

\int \frac{3 x ^{2}}{( x - 3 ) ( x ^{2} + 1 )} d x

Lösung

3 (\frac{9}{10} ln ∣ x - 3 ∣ + \frac{1}{10} (\frac{1}{2} ln x^{2} + 1 + 3 arctan (x))) + C

Schritte zur Lösung

\int \frac{3 x ^{2}}{( x - 3 ) ( x ^{2} + 1 )} d x

Entferne die Konstante:

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

= 3 \cdot \int \frac{x ^{2}}{( x - 3 ) ( x ^{2} + 1 )} d x

Ermittle den Partialbruch von

\frac{x ^{2}}{( x - 3 ) ( x ^{2} + 1 )} : \frac{9}{10 ( x - 3 )} + \frac{x + 3}{10 ( x ^{2} + 1 )}

= 3 \cdot \int \frac{9}{10 ( x - 3 )} + \frac{x + 3}{10 ( 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

= 3 (\int \frac{9}{10 ( x - 3 )} d x + \int \frac{x + 3}{10 ( x ^{2} + 1 )} d x)

\int \frac{9}{10 ( x - 3 )} d x = \frac{9}{10} ln ∣ x - 3 ∣

\int \frac{x + 3}{10 ( x ^{2} + 1 )} d x = \frac{1}{10} (\frac{1}{2} ln x^{2} + 1 + 3 arctan (x))

= 3 (\frac{9}{10} ln ∣ x - 3 ∣ + \frac{1}{10} (\frac{1}{2} ln x^{2} + 1 + 3 arctan (x)))

Füge eine Konstante zur Lösung hinzu

= 3 (\frac{9}{10} ln ∣ x - 3 ∣ + \frac{1}{10} (\frac{1}{2} ln x^{2} + 1 + 3 arctan (x))) + C

Graph

Beliebte Beispiele

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

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

integral of 2x(1+x^2)^8

\int 2 x (1 + x^{2})^{8} d x

integral of 3/(sqrt(5x-1))

\int \frac{3}{5 x - 1} d x

integral of xsin^2(x/2)

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

integral of x(sin(2x))

\int x (sin (2 x)) d x