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

The integral from 1 to 2 of 1/(2-3x) is -2/3 ln(2)

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

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

- \frac{2}{3} ln (2)

Decimal

- 0.46209 \dots

Solution steps

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

Apply u-substitution

: \int_{- 1}^{- 4} - \frac{1}{3 u} d u

= \int_{- 1}^{- 4} - \frac{1}{3 u} d u

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

= - \int_{- 4}^{- 1} - \frac{1}{3 u} d u

Take the constant out:

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

= - (- \frac{1}{3} \cdot \int_{- 4}^{- 1} \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - (- \frac{1}{3} [ln ∣ u ∣]_{- 4}^{- 1})

Simplify

= \frac{1}{3} [ln ∣ u ∣]_{- 4}^{- 1}

Compute the boundaries:

- 2 ln (2)

= \frac{1}{3} (- 2 ln (2))

Simplify

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

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