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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Factor

c^{2} - x^{2} : - (- c^{2} + x^{2})

= 2 \cdot \int \frac{1}{- ( - c ^{2} + x ^{2} )} d x

Take the constant out:

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

= 2 (- \int \frac{1}{- c ^{2} + x ^{2}} d x)

Apply Integral Substitution

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

Take the constant out:

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

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

Factor

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

= 2 (- \frac{1}{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

= 2 (- \frac{1}{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}

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

Substitute back

u = \frac{x}{c}

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

Simplify

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

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

Add a constant to the solution

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

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