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

The integral of 1/(a^2x^2-c^2) is -1/(2ac)(ln| a/c x+1|-ln| a/c x-1|)+C

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integral of 1/(a^2x^2-c^2)

\int \frac{1}{a ^{2} x ^{2} - c ^{2}} d x

- \frac{1}{2 a c} (ln \frac{a}{c} x + 1 - ln \frac{a}{c} x - 1) + C

Solution steps

\int \frac{1}{a ^{2} x ^{2} - c ^{2}} d x

Apply Integral Substitution

= \int \frac{1}{a c ( 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}{a c} \cdot \int \frac{1}{u ^{2} - 1} d u

Factor

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

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

Substitute back

u = \frac{a}{c} x

= \frac{1}{a c} (- (\frac{ln \frac{a}{c} x + 1}{2} - \frac{ln \frac{a}{c} x - 1}{2}))

Simplify

\frac{1}{a c} (- (\frac{ln \frac{a}{c} x + 1}{2} - \frac{ln \frac{a}{c} x - 1}{2})) : - \frac{1}{2 a c} (ln \frac{a}{c} x + 1 - ln \frac{a}{c} x - 1)

= - \frac{1}{2 a c} (ln \frac{a}{c} x + 1 - ln \frac{a}{c} x - 1)

Add a constant to the solution

= - \frac{1}{2 a c} (ln \frac{a}{c} x + 1 - ln \frac{a}{c} x - 1) + C

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What is the integral of 1/(a^2x^2-c^2) ?
The integral of 1/(a^2x^2-c^2) is -1/(2ac)(ln| a/c x+1|-ln| a/c x-1|)+C