What is the integral of (sqrt(2x))/(2x-4) ?

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

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integral of (sqrt(2x))/(2x-4)

\int \frac{2 x}{2 x - 4} d x

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

Solution steps

\int \frac{2 x}{2 x - 4} d x

Simplify

\frac{2 x}{2 x - 4} : \frac{x}{2 ( x - 2 )}

= \int \frac{x}{2 ( x - 2 )} d x

Take the constant out:

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

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

Apply u-substitution

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

Take the constant out:

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

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

Apply Integral Substitution

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

Take the constant out:

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

= \frac{1}{2} \cdot 22 \cdot \int \frac{v ^{2}}{v ^{2} - 1} d v

Factor

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

= \frac{1}{2} \cdot 22 \cdot \int \frac{v ^{2}}{- ( - v ^{2} + 1 )} d v

Take the constant out:

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

= \frac{1}{2} \cdot 22 (- \int \frac{v ^{2}}{- v ^{2} + 1} d v)

\frac{v ^{2}}{- v ^{2} + 1} = \frac{1}{- v ^{2} + 1} - 1

= \frac{1}{2} \cdot 22 (- \int \frac{1}{- v ^{2} + 1} - 1 d v)

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} \cdot 22 (- (\int \frac{1}{- v ^{2} + 1} d v - \int 1 d v))

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

\int 1 d v = v

= \frac{1}{2} \cdot 22 (- (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2} - v))

Substitute back

= \frac{1}{2} \cdot 22 - \frac{ln \frac{x}{2} + 1}{2} - \frac{ln \frac{x}{2} - 1}{2} - \frac{x}{2}

Simplify

\frac{1}{2} \cdot 22 - \frac{ln \frac{x}{2} + 1}{2} - \frac{ln \frac{x}{2} - 1}{2} - \frac{x}{2} : - ln \frac{x}{2} + 1 + ln \frac{x}{2} - 1 + 2 x

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

Add a constant to the solution

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

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