What is the integral of 1/(x^2-12) ?

The integral of 1/(x^2-12) is -1/(4sqrt(3))(ln| x/(2sqrt(3))+1|-ln| x/(2sqrt(3))-1|)+C

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integral of 1/(x^2-12)

\int \frac{1}{x ^{2} - 12} d x

- \frac{1}{4 3} (ln \frac{x}{2 3} + 1 - ln \frac{x}{2 3} - 1) + C

Solution steps

\int \frac{1}{x ^{2} - 12} d x

Apply Integral Substitution

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

Take the constant out:

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

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

Factor

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

= \frac{1}{2 3} \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

= \frac{1}{2 3} (- \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}

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

Substitute back

u = \frac{x}{2 3}

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

Simplify

\frac{1}{2 3} - \frac{ln \frac{x}{2 3} + 1}{2} - \frac{ln \frac{x}{2 3} - 1}{2} : - \frac{1}{4 3} (ln \frac{x}{2 3} + 1 - ln \frac{x}{2 3} - 1)

= - \frac{1}{4 3} (ln \frac{x}{2 3} + 1 - ln \frac{x}{2 3} - 1)

Add a constant to the solution

= - \frac{1}{4 3} (ln \frac{x}{2 3} + 1 - ln \frac{x}{2 3} - 1) + C

\int \frac{x + 4}{x ^{3} + 3 x ^{2} - 10 x} d x

\int - 3 π sin (π) x d x

f (x) = - 2 sin (x), at x = \frac{3}{4} π

x^{2} y^{^{'}} + \frac{4}{3} x y = - 2 cos (2 x) y^{- \frac{1}{2}}

n = 1 \sum \infty sin (nπ)

What is the integral of 1/(x^2-12) ?
The integral of 1/(x^2-12) is -1/(4sqrt(3))(ln| x/(2sqrt(3))+1|-ln| x/(2sqrt(3))-1|)+C