What is the integral of (3x^2+3x-23)/(x^2-9) ?

The integral of (3x^2+3x-23)/(x^2-9) is 3x-2/3 ln| x/3+1|+2/3 ln| x/3-1|+3/2 ln|x^2-9|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (3x^2+3x-23)/(x^2-9)

\int \frac{3 x ^{2} + 3 x - 23}{x ^{2} - 9} d x

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

Solution steps

\int \frac{3 x ^{2} + 3 x - 23}{x ^{2} - 9} d x

Expand

\frac{3 x ^{2} + 3 x - 23}{x ^{2} - 9} : \frac{3 x ^{2}}{x ^{2} - 9} + \frac{3 x}{x ^{2} - 9} - \frac{23}{x ^{2} - 9}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{3 x ^{2}}{x ^{2} - 9} d x + \int \frac{3 x}{x ^{2} - 9} d x - \int \frac{23}{x ^{2} - 9} d x

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

\int \frac{3 x}{x ^{2} - 9} d x = \frac{3}{2} ln x^{2} - 9

\int \frac{23}{x ^{2} - 9} d x = - \frac{23}{6} (ln \frac{x}{3} + 1 - ln \frac{x}{3} - 1)

= - 9 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1 - \frac{x}{3}) + \frac{3}{2} ln x^{2} - 9 - (- \frac{23}{6} (ln \frac{x}{3} + 1 - ln \frac{x}{3} - 1))

Simplify

- 9 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1 - \frac{x}{3}) + \frac{3}{2} ln x^{2} - 9 - (- \frac{23}{6} (ln \frac{x}{3} + 1 - ln \frac{x}{3} - 1)) : 3 x - \frac{2}{3} ln \frac{x}{3} + 1 + \frac{2}{3} ln \frac{x}{3} - 1 + \frac{3}{2} ln x^{2} - 9

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

Add a constant to the solution

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

\int \frac{4 - 9 x ^{2}}{x ^{2}} d x

\int \frac{3 x - 5}{x ^{4}} d x

\int \frac{x ^{2}}{324 - x ^{2}} d x

\int \frac{1}{( y - 1 ) ^{3}} d y

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

What is the integral of (3x^2+3x-23)/(x^2-9) ?
The integral of (3x^2+3x-23)/(x^2-9) is 3x-2/3 ln| x/3+1|+2/3 ln| x/3-1|+3/2 ln|x^2-9|+C