What is the integral of 1/(3-9x^2) ?

The integral of 1/(3-9x^2) is 1/(6sqrt(3))(ln|sqrt(3)x+1|-ln|sqrt(3)x-1|)+C

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integral of 1/(3-9x^2)

\int \frac{1}{3 - 9 x ^{2}} d x

\frac{1}{6 3} (ln 3 x + 1 - ln 3 x - 1) + C

Solution steps

\int \frac{1}{3 - 9 x ^{2}} d x

Apply Integral Substitution

= \int \frac{1}{3 ( 3 - 3 u ^{2} )} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{1}{3 - 3 u ^{2}} d u

\frac{1}{3 - 3 u ^{2}} = \frac{1}{3} \frac{1}{1 - u ^{2}}

= \frac{1}{3} \cdot \int \frac{1}{3} \cdot \frac{1}{1 - u ^{2}} d u

Take the constant out:

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

= \frac{1}{3} \cdot \frac{1}{3} \cdot \int \frac{1}{1 - u ^{2}} d u

Use the common integral:

\int \frac{1}{1 - u ^{2}} d u = \frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}

= \frac{1}{3} \cdot \frac{1}{3} (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2})

Substitute back

u = \frac{3}{3} x

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

Simplify

\frac{1}{3} \cdot \frac{1}{3} \frac{ln \frac{3}{3} x + 1}{2} - \frac{ln \frac{3}{3} x - 1}{2} : \frac{1}{6 3} (ln 3 x + 1 - ln 3 x - 1)

= \frac{1}{6 3} (ln 3 x + 1 - ln 3 x - 1)

Add a constant to the solution

= \frac{1}{6 3} (ln 3 x + 1 - ln 3 x - 1) + C

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What is the integral of 1/(3-9x^2) ?
The integral of 1/(3-9x^2) is 1/(6sqrt(3))(ln|sqrt(3)x+1|-ln|sqrt(3)x-1|)+C