What is the integral from 4 to 7 of-9log_{3}(x) ?

The integral from 4 to 7 of-9log_{3}(x) is -(9(7ln(7)-3-8ln(2)))/(ln(3))

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integral from 4 to 7 of-9log_{3}(x)

\int_{4}^{7} - 9 lo g_{3} (x) d x

- \frac{9 ( 7 ln ( 7 ) - 3 - 8 ln ( 2 ) )}{ln ( 3 )}

Decimal Notation

- 41.58495 \dots

Solution steps

\int_{4}^{7} - 9 lo g_{3} (x) d x

Take the constant out:

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

= - 9 \cdot \int_{4}^{7} lo g_{3} (x) d x

Apply Integration By Parts

: [x lo g_{3} (x) - \int \frac{1}{ln ( 3 )} d x]_{4}^{7}

= - 9 [x lo g_{3} (x) - \int \frac{1}{ln ( 3 )} d x]_{4}^{7}

\int \frac{1}{ln ( 3 )} d x = \frac{1}{ln ( 3 )} x

= - 9 [x lo g_{3} (x) - \frac{1}{ln ( 3 )} x]_{4}^{7}

Join

x lo g_{3} (x) - \frac{1}{ln ( 3 )} x : \frac{- x + x ln ( x )}{ln ( 3 )}

= - 9 [\frac{- x + x ln ( x )}{ln ( 3 )}]_{4}^{7}

Compute the boundaries:

\frac{7 ln ( 7 ) - 3 - 8 ln ( 2 )}{ln ( 3 )}

= - 9 \cdot \frac{7 ln ( 7 ) - 3 - 8 ln ( 2 )}{ln ( 3 )}

Simplify

= - \frac{9 ( 7 ln ( 7 ) - 3 - 8 ln ( 2 ) )}{ln ( 3 )}

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What is the integral from 4 to 7 of-9log_{3}(x) ?
The integral from 4 to 7 of-9log_{3}(x) is -(9(7ln(7)-3-8ln(2)))/(ln(3))