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

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

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integral of 1/(5+2x)

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

\frac{1}{2} ln ∣ 5 + 2 x ∣ + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{2 u} d u

= \int \frac{1}{2 u} d u

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} ln ∣ u ∣

Substitute back

u = 5 + 2 x

= \frac{1}{2} ln ∣ 5 + 2 x ∣

Add a constant to the solution

= \frac{1}{2} ln ∣ 5 + 2 x ∣ + C

(x^{2} + 1)^{(x + 1)}

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

f (x) = x arctan (x)

x \to 0 lim (\frac{1}{x ^{\frac{2}{3}}})

\frac{d}{d s} (ln (s^{2} + 1))