What is the integral from 0 to 8 of (56)/(7x+8) ?

The integral from 0 to 8 of (56)/(7x+8) is 24ln(2)

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integral from 0 to 8 of (56)/(7x+8)

\int_{0}^{8} \frac{56}{7 x + 8} d x

24 ln (2)

Decimal

16.63553 \dots

Solution steps

\int_{0}^{8} \frac{56}{7 x + 8} d x

Take the constant out:

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

= 56 \cdot \int_{0}^{8} \frac{1}{7 x + 8} d x

Apply u-substitution

: \int_{8}^{64} \frac{1}{7 u} d u

= 56 \cdot \int_{8}^{64} \frac{1}{7 u} d u

Take the constant out:

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

= 56 \cdot \frac{1}{7} \cdot \int_{8}^{64} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 56 \cdot \frac{1}{7} [ln ∣ u ∣]_{8}^{64}

Simplify

56 \cdot \frac{1}{7} [ln ∣ u ∣]_{8}^{64} : 8 [ln ∣ u ∣]_{8}^{64}

= 8 [ln ∣ u ∣]_{8}^{64}

Compute the boundaries:

3 ln (2)

= 8 \cdot 3 ln (2)

Simplify

= 24 ln (2)

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What is the integral from 0 to 8 of (56)/(7x+8) ?
The integral from 0 to 8 of (56)/(7x+8) is 24ln(2)