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

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

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

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

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

Decimal

2.01179 \dots

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int_{1}^{5} \frac{1}{4 u} d u

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

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

Simplify

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

= \frac{5}{4} [ln ∣ u ∣]_{1}^{5}

Compute the boundaries:

ln (5)

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

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