What is the integral of 1/(5x+3) ?

The integral of 1/(5x+3) is 1/5 ln|5x+3|+C

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integral of 1/(5x+3)

\int \frac{1}{5 x + 3} d x

\frac{1}{5} ln ∣ 5 x + 3 ∣ + C

Solution steps

\int \frac{1}{5 x + 3} 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 + 3

= \frac{1}{5} ln ∣ 5 x + 3 ∣

Add a constant to the solution

= \frac{1}{5} ln ∣ 5 x + 3 ∣ + C

\int \frac{1}{100 - y} d y

\frac{\partial}{\partial x} (e^{x - 2} - \frac{y}{2})

\frac{d}{d x} (\frac{sin ( x )}{3 x})

\int 1.5 x^{2} d x

\frac{d y}{d x} + y^{5} x + 8 y = 0