What is the integral from 0 to 2 of 4/(5x+1) ?

The integral from 0 to 2 of 4/(5x+1) is 4/5 ln(11)

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integral from 0 to 2 of 4/(5x+1)

\int_{0}^{2} \frac{4}{5 x + 1} d x

\frac{4}{5} ln (11)

Decimal

1.91831 \dots

Solution steps

\int_{0}^{2} \frac{4}{5 x + 1} d x

Take the constant out:

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

= 4 \cdot \int_{0}^{2} \frac{1}{5 x + 1} d x

Apply u-substitution

: \int_{1}^{11} \frac{1}{5 u} d u

= 4 \cdot \int_{1}^{11} \frac{1}{5 u} d u

Take the constant out:

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

= 4 \cdot \frac{1}{5} \cdot \int_{1}^{11} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 4 \cdot \frac{1}{5} [ln ∣ u ∣]_{1}^{11}

Simplify

4 \cdot \frac{1}{5} [ln ∣ u ∣]_{1}^{11} : \frac{4}{5} [ln ∣ u ∣]_{1}^{11}

= \frac{4}{5} [ln ∣ u ∣]_{1}^{11}

Compute the boundaries:

ln (11)

= \frac{4}{5} ln (11)

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What is the integral from 0 to 2 of 4/(5x+1) ?
The integral from 0 to 2 of 4/(5x+1) is 4/5 ln(11)