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

The integral of x/(5x-2) is 1/25 (5x-2+2ln|5x-2|)+C

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integral of x/(5x-2)

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

\frac{1}{25} (5 x - 2 + 2 ln ∣ 5 x - 2 ∣) + C

Solution steps

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

Apply u-substitution

: \int \frac{u + 2}{25 u} d u

= \int \frac{u + 2}{25 u} d u

Take the constant out:

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

= \frac{1}{25} \cdot \int \frac{u + 2}{u} d u

Expand

\frac{u + 2}{u} : 1 + \frac{2}{u}

= \frac{1}{25} \cdot \int 1 + \frac{2}{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}{25} (\int 1 d u + \int \frac{2}{u} d u)

\int 1 d u = u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= \frac{1}{25} (u + 2 ln ∣ u ∣)

Substitute back

u = 5 x - 2

= \frac{1}{25} (5 x - 2 + 2 ln ∣ 5 x - 2 ∣)

Add a constant to the solution

= \frac{1}{25} (5 x - 2 + 2 ln ∣ 5 x - 2 ∣) + C

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\frac{d}{d x} (\frac{4 x ^{3} + 2 x + 5}{x})

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What is the integral of x/(5x-2) ?
The integral of x/(5x-2) is 1/25 (5x-2+2ln|5x-2|)+C