What is the integral of 1/((x-2)(sqrt(x+2))) ?

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

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integral of 1/((x-2)(sqrt(x+2)))

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

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

Solution steps

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

Apply u-substitution

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

Take the constant out:

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

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

Apply Integral Substitution

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

Take the constant out:

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

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

Factor

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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