What is the integral from 0 to 5 of (5+3x)^{-1} ?

The integral from 0 to 5 of (5+3x)^{-1} is 2/3 ln(2)

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integral from 0 to 5 of (5+3x)^{-1}

\int_{0}^{5} (5 + 3 x)^{- 1} d x

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

Decimal

0.46209 \dots

Solution steps

\int_{0}^{5} (5 + 3 x)^{- 1} d x

Apply exponent rule:

a^{- 1} = \frac{1}{a}

= \int_{0}^{5} \frac{1}{5 + 3 x} d x

Apply u-substitution

= \int_{5}^{20} \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_{5}^{20} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{3} [ln ∣ u ∣]_{5}^{20}

Compute the boundaries:

ln (20) - ln (5)

= \frac{1}{3} (ln (20) - ln (5))

Simplify

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

What is the integral from 0 to 5 of (5+3x)^{-1} ?
The integral from 0 to 5 of (5+3x)^{-1} is 2/3 ln(2)