What is the integral from 8 to 12 of ln(x/2-3) ?

The integral from 8 to 12 of ln(x/2-3) is 6ln(3)-4

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integral from 8 to 12 of ln(x/2-3)

\int_{8}^{12} ln (\frac{x}{2} - 3) d x

6 ln (3) - 4

Decimal

2.59167 \dots

Solution steps

\int_{8}^{12} ln (\frac{x}{2} - 3) d x

Apply Integration By Parts

= [x ln (\frac{x}{2} - 3) - \int \frac{x}{x - 6} d x]_{8}^{12}

\int \frac{x}{x - 6} d x = x - 6 + 6 ln ∣ x - 6 ∣

= [x ln (\frac{x}{2} - 3) - (x - 6 + 6 ln ∣ x - 6 ∣)]_{8}^{12}

Simplify

= [x ln (\frac{x}{2} - 3) - x + 6 - 6 ln ∣ x - 6 ∣]_{8}^{12}

Compute the boundaries:

12 ln (3) + 6 ln (2) - 4 - 6 ln (6)

= 12 ln (3) + 6 ln (2) - 4 - 6 ln (6)

Simplify

= 6 ln (3) - 4

What is the integral from 8 to 12 of ln(x/2-3) ?
The integral from 8 to 12 of ln(x/2-3) is 6ln(3)-4