What is the integral of 4/((x-sqrt(3))(x+\sqrt{3))} ?

The integral of 4/((x-sqrt(3))(x+\sqrt{3))} is -2/(sqrt(3))(ln| x/(sqrt(3))+1|-ln| x/(sqrt(3))-1|)+C

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integral of 4/((x-sqrt(3))(x+\sqrt{3))}

\int \frac{4}{( x - 3 ) ( x + 3 )} d x

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

Solution steps

\int \frac{4}{( x - 3 ) ( x + 3 )} d x

Take the constant out:

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

= 4 \cdot \int \frac{1}{( x - 3 ) ( x + 3 )} d x

Apply Integral Substitution

= 4 \cdot \int \frac{1}{3 ( u - 1 ) ( u + 1 )} d u

Take the constant out:

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

= 4 \cdot \frac{1}{3} \cdot \int \frac{1}{( u - 1 ) ( u + 1 )} d u

Expand

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

= 4 \cdot \frac{1}{3} \cdot \int \frac{1}{u ^{2} - 1} d u

Factor

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

= 4 \cdot \frac{1}{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

= 4 \cdot \frac{1}{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}

= 4 \cdot \frac{1}{3} (- (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}))

Substitute back

u = \frac{x}{3}

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

Simplify

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

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

Add a constant to the solution

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

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