What is the integral from 2 to 8 of 5/(4-3x) ?

The integral from 2 to 8 of 5/(4-3x) is -5/3 ln(10)

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integral from 2 to 8 of 5/(4-3x)

\int_{2}^{8} \frac{5}{4 - 3 x} d x

- \frac{5}{3} ln (10)

Decimal

- 3.83764 \dots

Solution steps

\int_{2}^{8} \frac{5}{4 - 3 x} d x

Take the constant out:

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

= 5 \cdot \int_{2}^{8} \frac{1}{4 - 3 x} d x

Apply u-substitution

= 5 \cdot \int_{- 2}^{- 20} - \frac{1}{3 u} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= 5 (- \int_{- 20}^{- 2} - \frac{1}{3 u} d u)

Take the constant out:

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

= 5 (- (- \frac{1}{3} \cdot \int_{- 20}^{- 2} \frac{1}{u} d u))

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 (- (- \frac{1}{3} [ln ∣ u ∣]_{- 20}^{- 2}))

Simplify

5 (- (- \frac{1}{3} [ln ∣ u ∣]_{- 20}^{- 2})) : \frac{5}{3} [ln ∣ u ∣]_{- 20}^{- 2}

= \frac{5}{3} [ln ∣ u ∣]_{- 20}^{- 2}

Compute the boundaries:

ln (2) - ln (20)

= \frac{5}{3} (ln (2) - ln (20))

Simplify

= - \frac{5}{3} ln (10)

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What is the integral from 2 to 8 of 5/(4-3x) ?
The integral from 2 to 8 of 5/(4-3x) is -5/3 ln(10)