What is the integral of 1/(5x+2) ?

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

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integral of 1/(5x+2)

\int \frac{1}{5 x + 2} d x

\frac{1}{5} ln ∣ 5 x + 2 ∣ + C

Solution steps

\int \frac{1}{5 x + 2} d x

Apply u-substitution

: \int \frac{1}{5 u} d u

= \int \frac{1}{5 u} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{5} ln ∣ u ∣

Substitute back

u = 5 x + 2

= \frac{1}{5} ln ∣ 5 x + 2 ∣

Add a constant to the solution

= \frac{1}{5} ln ∣ 5 x + 2 ∣ + C

\int_{1}^{2} x e^{3 x^{2}} d x

\frac{d y}{d x} = \frac{y}{x}

\frac{\partial}{\partial y} (\frac{cy}{x ^{2} + y ^{2}})

(4 + x) y^{^{'}} = y^{3}

x \to 2 lim (\frac{1}{x - 1})