What is the integral of x/(5x+1) ?

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

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integral of x/(5x+1)

\int \frac{x}{5 x + 1} d x

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

Solution steps

\int \frac{x}{5 x + 1} d x

Apply u-substitution

: \int \frac{u - 1}{25 u} d u

= \int \frac{u - 1}{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 - 1}{u} d u

Expand

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

= \frac{1}{25} \cdot \int 1 - \frac{1}{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{1}{u} d u)

\int 1 d u = u

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{1}{25} (u - ln ∣ u ∣)

Substitute back

u = 5 x + 1

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

Add a constant to the solution

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

2 x + 7

(2 x y^{2} - 5) d x + (2 x^{2} y + 4) d y = 0

x \to 1 lim (x^{4} - 1)

g (x) = 3 - x

\frac{\partial}{\partial x} (4 x + 16 y)

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