What is the integral from 2 to 7 of 5log_{6}(x) ?

The integral from 2 to 7 of 5log_{6}(x) is (5(7ln(7)-5-2ln(2)))/(ln(6))

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integral from 2 to 7 of 5log_{6}(x)

\int_{2}^{7} 5 lo g_{6} (x) d x

\frac{5 ( 7 ln ( 7 ) - 5 - 2 ln ( 2 ) )}{ln ( 6 )}

Decimal

20.18986 \dots

Solution steps

\int_{2}^{7} 5 lo g_{6} (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}^{7} lo g_{6} (x) d x

Apply Integration By Parts

: [x lo g_{6} (x) - \int \frac{1}{ln ( 6 )} d x]_{2}^{7}

= 5 [x lo g_{6} (x) - \int \frac{1}{ln ( 6 )} d x]_{2}^{7}

\int \frac{1}{ln ( 6 )} d x = \frac{1}{ln ( 6 )} x

= 5 [x lo g_{6} (x) - \frac{1}{ln ( 6 )} x]_{2}^{7}

Join

x lo g_{6} (x) - \frac{1}{ln ( 6 )} x : \frac{- x + x ln ( x )}{ln ( 6 )}

= 5 [\frac{- x + x ln ( x )}{ln ( 6 )}]_{2}^{7}

Compute the boundaries:

\frac{7 ln ( 7 ) - 5 - 2 ln ( 2 )}{ln ( 6 )}

= 5 \cdot \frac{7 ln ( 7 ) - 5 - 2 ln ( 2 )}{ln ( 6 )}

Simplify

= \frac{5 ( 7 ln ( 7 ) - 5 - 2 ln ( 2 ) )}{ln ( 6 )}

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What is the integral from 2 to 7 of 5log_{6}(x) ?
The integral from 2 to 7 of 5log_{6}(x) is (5(7ln(7)-5-2ln(2)))/(ln(6))