What is the integral of 9/(x^2-81) ?

The integral of 9/(x^2-81) is -1/2 ln| x/9+1|+1/2 ln| x/9-1|+C

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integral of 9/(x^2-81)

\int \frac{9}{x ^{2} - 81} d x

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

Solution steps

\int \frac{9}{x ^{2} - 81} d x

Take the constant out:

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

= 9 \cdot \int \frac{1}{x ^{2} - 81} d x

Apply Integral Substitution

= 9 \cdot \int \frac{1}{9 ( u ^{2} - 1 )} d u

Take the constant out:

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

= 9 \cdot \frac{1}{9} \cdot \int \frac{1}{u ^{2} - 1} d u

Factor

u^{2} - 1 : - (- u^{2} + 1)

= 9 \cdot \frac{1}{9} \cdot \int \frac{1}{- ( - u ^{2} + 1 )} d u

Take the constant out:

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

= 9 \cdot \frac{1}{9} (- \int \frac{1}{- u ^{2} + 1} d u)

Use the common integral:

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

= 9 \cdot \frac{1}{9} (- (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}))

Substitute back

u = \frac{x}{9}

= 9 \cdot \frac{1}{9} (- (\frac{ln \frac{x}{9} + 1}{2} - \frac{ln \frac{x}{9} - 1}{2}))

Simplify

9 \cdot \frac{1}{9} (- (\frac{ln \frac{x}{9} + 1}{2} - \frac{ln \frac{x}{9} - 1}{2})) : - \frac{1}{2} ln \frac{x}{9} + 1 + \frac{1}{2} ln \frac{x}{9} - 1

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

Add a constant to the solution

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

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What is the integral of 9/(x^2-81) ?
The integral of 9/(x^2-81) is -1/2 ln| x/9+1|+1/2 ln| x/9-1|+C