What is the integral from 3 to 12 of log_{10}(x-2) ?

The integral from 3 to 12 of log_{10}(x-2) is (10ln(10)-9)/(ln(10))

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integral from 3 to 12 of log_{10}(x-2)

\int_{3}^{12} lo g_{10} (x - 2) d x

\frac{10 ln ( 10 ) - 9}{ln ( 10 )}

Decimal

6.09134 \dots

Solution steps

\int_{3}^{12} lo g_{10} (x - 2) d x

Apply Integration By Parts

: [x lo g_{10} (x - 2) - \int \frac{x}{ln ( 10 ) ( x - 2 )} d x]_{3}^{12}

= [x lo g_{10} (x - 2) - \int \frac{x}{ln ( 10 ) ( x - 2 )} d x]_{3}^{12}

\int \frac{x}{ln ( 10 ) ( x - 2 )} d x = \frac{1}{ln ( 10 )} (x - 2 + 2 ln ∣ x - 2 ∣)

= [x lo g_{10} (x - 2) - \frac{1}{ln ( 10 )} (x - 2 + 2 ln ∣ x - 2 ∣)]_{3}^{12}

Join

x lo g_{10} (x - 2) - \frac{1}{ln ( 10 )} (x - 2 + 2 ln ∣ x - 2 ∣) : \frac{- x + 2 - 2 ln ∣ x - 2 ∣ + x ln ( x - 2 )}{ln ( 10 )}

= [\frac{- x + 2 - 2 ln ∣ x - 2 ∣ + x ln ( x - 2 )}{ln ( 10 )}]_{3}^{12}

Compute the boundaries:

\frac{10 ln ( 10 ) - 9}{ln ( 10 )}

= \frac{10 ln ( 10 ) - 9}{ln ( 10 )}

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What is the integral from 3 to 12 of log_{10}(x-2) ?
The integral from 3 to 12 of log_{10}(x-2) is (10ln(10)-9)/(ln(10))