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

The integral of x/(2x+5) is 1/4 (2x+5-5ln|2x+5|)+C

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integral of x/(2x+5)

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

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

Solution steps

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

Apply u-substitution

: \int \frac{u - 5}{4 u} d u

= \int \frac{u - 5}{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 - 5}{u} d u

Expand

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

= \frac{1}{4} \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

= \frac{1}{4} (\int 1 d u - \int \frac{5}{u} d u)

\int 1 d u = u

\int \frac{5}{u} d u = 5 ln ∣ u ∣

= \frac{1}{4} (u - 5 ln ∣ u ∣)

Substitute back

u = 2 x + 5

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

Add a constant to the solution

= \frac{1}{4} (2 x + 5 - 5 ln ∣ 2 x + 5 ∣) + C

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What is the integral of x/(2x+5) ?
The integral of x/(2x+5) is 1/4 (2x+5-5ln|2x+5|)+C