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

The integral from 1 to 3 of 2/(5+3x) is 2/3 ln(7/4)

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integral from 1 to 3 of 2/(5+3x)

\int_{1}^{3} \frac{2}{5 + 3 x} d x

\frac{2}{3} ln (\frac{7}{4})

Decimal

0.37307 \dots

Solution steps

\int_{1}^{3} \frac{2}{5 + 3 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}{5 + 3 x} d x

Apply u-substitution

= 2 \cdot \int_{8}^{14} \frac{1}{3 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}{3} \cdot \int_{8}^{14} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{3} [ln ∣ u ∣]_{8}^{14}

Simplify

2 \cdot \frac{1}{3} [ln ∣ u ∣]_{8}^{14} : \frac{2}{3} [ln ∣ u ∣]_{8}^{14}

= \frac{2}{3} [ln ∣ u ∣]_{8}^{14}

Compute the boundaries:

ln (14) - 3 ln (2)

= \frac{2}{3} (ln (14) - 3 ln (2))

Simplify

= \frac{2}{3} ln (\frac{7}{4})

What is the integral from 1 to 3 of 2/(5+3x) ?
The integral from 1 to 3 of 2/(5+3x) is 2/3 ln(7/4)