What is the integral of (4x)/(x-8) ?

The integral of (4x)/(x-8) is 4(x-8+8ln|x-8|)+C

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integral of (4x)/(x-8)

\int \frac{4 x}{x - 8} d x

4 (x - 8 + 8 ln ∣ x - 8 ∣) + C

Solution steps

\int \frac{4 x}{x - 8} d x

Take the constant out:

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

= 4 \cdot \int \frac{x}{x - 8} d x

Apply u-substitution

: \int \frac{u + 8}{u} d u

= 4 \cdot \int \frac{u + 8}{u} d u

Expand

\frac{u + 8}{u} : 1 + \frac{8}{u}

= 4 \cdot \int 1 + \frac{8}{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

= 4 (\int 1 d u + \int \frac{8}{u} d u)

\int 1 d u = u

\int \frac{8}{u} d u = 8 ln ∣ u ∣

= 4 (u + 8 ln ∣ u ∣)

Substitute back

u = x - 8

= 4 (x - 8 + 8 ln ∣ x - 8 ∣)

Add a constant to the solution

= 4 (x - 8 + 8 ln ∣ x - 8 ∣) + C

x \to 0 lim (arcsin (1 + 2 x))

\int_{1}^{5} \frac{7}{x ^{2}} d x

\frac{d}{d x} ((5 x + 8)^{3})

\frac{d}{d x} ((\frac{x - 7}{x + 2})^{2})

x^{\frac{1}{3}}, 1000

What is the integral of (4x)/(x-8) ?
The integral of (4x)/(x-8) is 4(x-8+8ln|x-8|)+C