integral of (6x^3+2x^2-45x-15)/(x^2-9)

Lösung

\int \frac{6 x ^{3} + 2 x ^{2} - 45 x - 15}{x ^{2} - 9} d x

3 x^{2} + 2 x - \frac{1}{2} ln \frac{x}{3} + 1 + \frac{1}{2} ln \frac{x}{3} - 1 + \frac{9}{2} ln x^{2} - 9 - 27 + C

Schritte zur Lösung

\int \frac{6 x ^{3} + 2 x ^{2} - 45 x - 15}{x ^{2} - 9} d x

Multipliziere aus

\frac{6 x ^{3} + 2 x ^{2} - 45 x - 15}{x ^{2} - 9} : \frac{6 x ^{3}}{x ^{2} - 9} + \frac{2 x ^{2}}{x ^{2} - 9} - \frac{45 x}{x ^{2} - 9} - \frac{15}{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{6 x ^{3}}{x ^{2} - 9} d x + \int \frac{2 x ^{2}}{x ^{2} - 9} d x - \int \frac{45 x}{x ^{2} - 9} d x - \int \frac{15}{x ^{2} - 9} d x

\int \frac{6 x ^{3}}{x ^{2} - 9} d x = 3 (x^{2} - 9 + 9 ln x^{2} - 9)

\int \frac{2 x ^{2}}{x ^{2} - 9} d x = - 6 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1 - \frac{x}{3})

\int \frac{45 x}{x ^{2} - 9} d x = \frac{45}{2} ln x^{2} - 9

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

= 3 (x^{2} - 9 + 9 ln x^{2} - 9) - 6 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1 - \frac{x}{3}) - \frac{45}{2} ln x^{2} - 9 - (- 5 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1))

Vereinfache

3 (x^{2} - 9 + 9 ln x^{2} - 9) - 6 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1 - \frac{x}{3}) - \frac{45}{2} ln x^{2} - 9 - (- 5 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1)) : 3 x^{2} + 2 x - \frac{1}{2} ln \frac{x}{3} + 1 + \frac{1}{2} ln \frac{x}{3} - 1 + \frac{9}{2} ln x^{2} - 9 - 27

= 3 x^{2} + 2 x - \frac{1}{2} ln \frac{x}{3} + 1 + \frac{1}{2} ln \frac{x}{3} - 1 + \frac{9}{2} ln x^{2} - 9 - 27

Füge eine Konstante zur Lösung hinzu

= 3 x^{2} + 2 x - \frac{1}{2} ln \frac{x}{3} + 1 + \frac{1}{2} ln \frac{x}{3} - 1 + \frac{9}{2} ln x^{2} - 9 - 27 + C

d er i v a t i v e 6 - x

n = 0 \sum \infty \frac{n ^{2}}{2 ^{n}}

\frac{d}{d x} (x sin (\frac{6}{x}))

a re a y = \frac{x ^{2}}{2} - 3, y = - \frac{x ^{2}}{2} + 2 x

\int_{1}^{9} \frac{1}{x - 7} d x