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

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

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

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

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

Decimal

0.55111 \dots

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int_{7}^{16} \frac{1}{3 u} d u

= 2 \cdot \int_{7}^{16} \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_{7}^{16} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{3} [ln ∣ u ∣]_{7}^{16}

Simplify

2 \cdot \frac{1}{3} [ln ∣ u ∣]_{7}^{16} : \frac{2}{3} [ln ∣ u ∣]_{7}^{16}

= \frac{2}{3} [ln ∣ u ∣]_{7}^{16}

Compute the boundaries:

4 ln (2) - ln (7)

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

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