What is the integral from 5 to 6 of 2log_{10}(x) ?

The integral from 5 to 6 of 2log_{10}(x) is (2(6ln(6)-1-5ln(5)))/(ln(10))

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integral from 5 to 6 of 2log_{10}(x)

\int_{5}^{6} 2 lo g_{10} (x) d x

\frac{2 ( 6 ln ( 6 ) - 1 - 5 ln ( 5 ) )}{ln ( 10 )}

Decimal

1.47952 \dots

Solution steps

\int_{5}^{6} 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_{5}^{6} lo g_{10} (x) d x

Apply Integration By Parts

: [x lo g_{10} (x) - \int \frac{1}{ln ( 10 )} d x]_{5}^{6}

= 2 [x lo g_{10} (x) - \int \frac{1}{ln ( 10 )} d x]_{5}^{6}

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

= 2 [x lo g_{10} (x) - \frac{1}{ln ( 10 )} x]_{5}^{6}

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 )}]_{5}^{6}

Compute the boundaries:

\frac{6 ln ( 6 ) - 1 - 5 ln ( 5 )}{ln ( 10 )}

= 2 \cdot \frac{6 ln ( 6 ) - 1 - 5 ln ( 5 )}{ln ( 10 )}

Simplify

= \frac{2 ( 6 ln ( 6 ) - 1 - 5 ln ( 5 ) )}{ln ( 10 )}

\int_{1}^{10} x^{2} - x d x

\int_{0}^{9} 8 t d t

\int_{0}^{1} ln (x) x d x

\int_{2}^{6} 7 d x

\int_{1}^{5} \frac{ln ( r )}{r ^{2}} d r

What is the integral from 5 to 6 of 2log_{10}(x) ?
The integral from 5 to 6 of 2log_{10}(x) is (2(6ln(6)-1-5ln(5)))/(ln(10))