What is the integral of (2x+1)/(3x^2-27) ?

The integral of (2x+1)/(3x^2-27) is 1/3 ln|3x^2-27|-1/18 (ln| x/3+1|-ln| x/3-1|)+C

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integral of (2x+1)/(3x^2-27)

\int \frac{2 x + 1}{3 x ^{2} - 27} d x

\frac{1}{3} ln 3 x^{2} - 27 - \frac{1}{18} (ln \frac{x}{3} + 1 - ln \frac{x}{3} - 1) + C

Solution steps

\int \frac{2 x + 1}{3 x ^{2} - 27} d x

Expand

\frac{2 x + 1}{3 x ^{2} - 27} : \frac{2 x}{3 x ^{2} - 27} + \frac{1}{3 x ^{2} - 27}

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{2 x}{3 x ^{2} - 27} d x + \int \frac{1}{3 x ^{2} - 27} d x

\int \frac{2 x}{3 x ^{2} - 27} d x = \frac{1}{3} ln 3 x^{2} - 27

\int \frac{1}{3 x ^{2} - 27} d x = - \frac{1}{18} (ln \frac{x}{3} + 1 - ln \frac{x}{3} - 1)

= \frac{1}{3} ln 3 x^{2} - 27 - \frac{1}{18} (ln \frac{x}{3} + 1 - ln \frac{x}{3} - 1)

Add a constant to the solution

= \frac{1}{3} ln 3 x^{2} - 27 - \frac{1}{18} (ln \frac{x}{3} + 1 - ln \frac{x}{3} - 1) + C

\int \frac{1}{e ^{- x} + 1} d x

\int \frac{1}{2 + 7 y ^{2}} d y

e^{- 2 t}

\frac{d}{d x} (\frac{5}{x + 4})

\int - 4 e^{- 2 x} d x

What is the integral of (2x+1)/(3x^2-27) ?
The integral of (2x+1)/(3x^2-27) is 1/3 ln|3x^2-27|-1/18 (ln| x/3+1|-ln| x/3-1|)+C