What is the integral from 1 to 3 of 2/(8+4x) ?

The integral from 1 to 3 of 2/(8+4x) is 1/2 ln(5/3)

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integral from 1 to 3 of 2/(8+4x)

\int_{1}^{3} \frac{2}{8 + 4 x} d x

\frac{1}{2} ln (\frac{5}{3})

Solution steps

\int_{1}^{3} \frac{2}{8 + 4 x} d x

Take the constant out:

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

= 2 \cdot \int_{1}^{3} \frac{1}{8 + 4 x} d x

Apply u-substitution

: \int_{12}^{20} \frac{1}{4 u} d u

= 2 \cdot \int_{12}^{20} \frac{1}{4 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}{4} \cdot \int_{12}^{20} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{4} [ln ∣ u ∣]_{12}^{20}

Simplify

2 \cdot \frac{1}{4} [ln ∣ u ∣]_{12}^{20} : \frac{1}{2} [ln ∣ u ∣]_{12}^{20}

= \frac{1}{2} [ln ∣ u ∣]_{12}^{20}

Compute the boundaries:

ln (20) - ln (12)

= \frac{1}{2} (ln (20) - ln (12))

Simplify

= \frac{1}{2} ln (\frac{5}{3})

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What is the integral from 1 to 3 of 2/(8+4x) ?
The integral from 1 to 3 of 2/(8+4x) is 1/2 ln(5/3)