What is the integral of (2x)/(x-5) ?

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

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integral of (2x)/(x-5)

\int \frac{2 x}{x - 5} d x

2 (x - 5 + 5 ln ∣ x - 5 ∣) + C

Solution steps

\int \frac{2 x}{x - 5} d x

Take the constant out:

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

= 2 \cdot \int \frac{x}{x - 5} d x

Apply u-substitution

: \int \frac{u + 5}{u} d u

= 2 \cdot \int \frac{u + 5}{u} d u

Expand

\frac{u + 5}{u} : 1 + \frac{5}{u}

= 2 \cdot \int 1 + \frac{5}{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

= 2 (\int 1 d u + \int \frac{5}{u} d u)

\int 1 d u = u

\int \frac{5}{u} d u = 5 ln ∣ u ∣

= 2 (u + 5 ln ∣ u ∣)

Substitute back

u = x - 5

= 2 (x - 5 + 5 ln ∣ x - 5 ∣)

Add a constant to the solution

= 2 (x - 5 + 5 ln ∣ x - 5 ∣) + C

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