What is the integral from 0 to 3 of (-2)/(5x+1) ?

The integral from 0 to 3 of (-2)/(5x+1) is -8/5 ln(2)

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integral from 0 to 3 of (-2)/(5x+1)

\int_{0}^{3} \frac{- 2}{5 x + 1} d x

- \frac{8}{5} ln (2)

Decimal

- 1.10903 \dots

Solution steps

\int_{0}^{3} \frac{- 2}{5 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_{0}^{3} \frac{1}{5 x + 1} d x

Apply u-substitution

= - 2 \cdot \int_{1}^{16} \frac{1}{5 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}{5} \cdot \int_{1}^{16} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - 2 \cdot \frac{1}{5} [ln ∣ u ∣]_{1}^{16}

Simplify

- 2 \cdot \frac{1}{5} [ln ∣ u ∣]_{1}^{16} : - \frac{2}{5} [ln ∣ u ∣]_{1}^{16}

= - \frac{2}{5} [ln ∣ u ∣]_{1}^{16}

Compute the boundaries:

4 ln (2)

= - \frac{2}{5} \cdot 4 ln (2)

Simplify

= - \frac{8}{5} ln (2)

What is the integral from 0 to 3 of (-2)/(5x+1) ?
The integral from 0 to 3 of (-2)/(5x+1) is -8/5 ln(2)