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

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

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integral of 8/(2+8x)

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

ln ∣ 2 + 8 x ∣ + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{1}{8 u} d u

= 8 \cdot \int \frac{1}{8 u} d u

Take the constant out:

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

= 8 \cdot \frac{1}{8} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 8 \cdot \frac{1}{8} ln ∣ u ∣

Substitute back

u = 2 + 8 x

= 8 \cdot \frac{1}{8} ln ∣ 2 + 8 x ∣

Simplify

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

= ln ∣ 2 + 8 x ∣

Add a constant to the solution

= ln ∣ 2 + 8 x ∣ + C

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