What is the integral of 2/(8+2x) ?

The integral of 2/(8+2x) is ln|8+2x|+C

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integral of 2/(8+2x)

\int \frac{2}{8 + 2 x} d x

ln ∣ 8 + 2 x ∣ + C

Solution steps

\int \frac{2}{8 + 2 x} d x

Take the constant out:

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

= 2 \cdot \int \frac{1}{8 + 2 x} d x

Apply u-substitution

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

= 2 \cdot \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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = 8 + 2 x

= 2 \cdot \frac{1}{2} ln ∣ 8 + 2 x ∣

Simplify

2 \cdot \frac{1}{2} ln ∣ 8 + 2 x ∣ : ln ∣ 8 + 2 x ∣

= ln ∣ 8 + 2 x ∣

Add a constant to the solution

= ln ∣ 8 + 2 x ∣ + C

\frac{d}{d x} (\frac{1}{2 x + 6})

\int t 2^{t + 2} d t

n \to \infty lim (\frac{1 n}{3})

8

x \to 0 + lim (x - sin (x))