What is the integral from 0 to 4 of 2/(3x+4) ?

The integral from 0 to 4 of 2/(3x+4) is 4/3 ln(2)

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integral from 0 to 4 of 2/(3x+4)

\int_{0}^{4} \frac{2}{3 x + 4} d x

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

Decimal

0.92419 \dots

Solution steps

\int_{0}^{4} \frac{2}{3 x + 4} d x

Take the constant out:

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

= 2 \cdot \int_{0}^{4} \frac{1}{3 x + 4} d x

Apply u-substitution

: \int_{4}^{16} \frac{1}{3 u} d u

= 2 \cdot \int_{4}^{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_{4}^{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 ∣]_{4}^{16}

Simplify

2 \cdot \frac{1}{3} [ln ∣ u ∣]_{4}^{16} : \frac{2}{3} [ln ∣ u ∣]_{4}^{16}

= \frac{2}{3} [ln ∣ u ∣]_{4}^{16}

Compute the boundaries:

2 ln (2)

= \frac{2}{3} \cdot 2 ln (2)

Simplify

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

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What is the integral from 0 to 4 of 2/(3x+4) ?
The integral from 0 to 4 of 2/(3x+4) is 4/3 ln(2)