What is the integral of x/(2x-10) ?

The integral of x/(2x-10) is 1/2 (x-5+5ln(2|x-5|))+C

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integral of x/(2x-10)

\int \frac{x}{2 x - 10} d x

\frac{1}{2} (x - 5 + 5 ln (2 ∣ x - 5 ∣)) + C

Solution steps

\int \frac{x}{2 x - 10} d x

Apply u-substitution

: \int \frac{u + 10}{4 u} d u

= \int \frac{u + 10}{4 u} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{u + 10}{u} d u

Expand

\frac{u + 10}{u} : 1 + \frac{10}{u}

= \frac{1}{4} \cdot \int 1 + \frac{10}{u} d u

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}{4} (\int 1 d u + \int \frac{10}{u} d u)

\int 1 d u = u

\int \frac{10}{u} d u = 10 ln ∣ u ∣

= \frac{1}{4} (u + 10 ln ∣ u ∣)

Substitute back

u = 2 x - 10

= \frac{1}{4} (2 x - 10 + 10 ln ∣ 2 x - 10 ∣)

Simplify

\frac{1}{4} (2 x - 10 + 10 ln ∣ 2 x - 10 ∣) : \frac{1}{2} (x - 5 + 5 ln (2 ∣ x - 5 ∣))

= \frac{1}{2} (x - 5 + 5 ln (2 ∣ x - 5 ∣))

Add a constant to the solution

= \frac{1}{2} (x - 5 + 5 ln (2 ∣ x - 5 ∣)) + C

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What is the integral of x/(2x-10) ?
The integral of x/(2x-10) is 1/2 (x-5+5ln(2|x-5|))+C