What is the integral from 3 to 7 of-2log_{10}(x) ?

The integral from 3 to 7 of-2log_{10}(x) is -(2(7ln(7)-4-3ln(3)))/(ln(10))

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integral from 3 to 7 of-2log_{10}(x)

\int_{3}^{7} - 2 lo g_{10} (x) d x

- \frac{2 ( 7 ln ( 7 ) - 4 - 3 ln ( 3 ) )}{ln ( 10 )}

Decimal

- 5.49428 \dots

Solution steps

\int_{3}^{7} - 2 lo g_{10} (x) d x

Take the constant out:

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

= - 2 \cdot \int_{3}^{7} lo g_{10} (x) d x

Apply Integration By Parts

= - 2 [x lo g_{10} (x) - \int \frac{1}{ln ( 10 )} d x]_{3}^{7}

\int \frac{1}{ln ( 10 )} d x = \frac{1}{ln ( 10 )} x

= - 2 [x lo g_{10} (x) - \frac{1}{ln ( 10 )} x]_{3}^{7}

Join

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

= - 2 [\frac{- x + x ln ( x )}{ln ( 10 )}]_{3}^{7}

Compute the boundaries:

\frac{7 ln ( 7 ) - 4 - 3 ln ( 3 )}{ln ( 10 )}

= - 2 \cdot \frac{7 ln ( 7 ) - 4 - 3 ln ( 3 )}{ln ( 10 )}

Simplify

= - \frac{2 ( 7 ln ( 7 ) - 4 - 3 ln ( 3 ) )}{ln ( 10 )}

What is the integral from 3 to 7 of-2log_{10}(x) ?
The integral from 3 to 7 of-2log_{10}(x) is -(2(7ln(7)-4-3ln(3)))/(ln(10))