What is the integral from 1 to 10 of ((ln(x))/(ln(10))) ?

The integral from 1 to 10 of ((ln(x))/(ln(10))) is (10ln(10)-9)/(ln(10))

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integral from 1 to 10 of ((ln(x))/(ln(10)))

\int_{1}^{10} (\frac{ln ( x )}{ln ( 10 )}) d x

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

Decimal

6.09134 \dots

Solution steps

\int_{1}^{10} \frac{ln ( x )}{ln ( 10 )} d x

Take the constant out:

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

= \frac{1}{ln ( 10 )} \cdot \int_{1}^{10} ln (x) d x

Apply Integration By Parts

: [x ln (x) - \int 1 d x]_{1}^{10}

= \frac{1}{ln ( 10 )} [x ln (x) - \int 1 d x]_{1}^{10}

\int 1 d x = x

= \frac{1}{ln ( 10 )} [x ln (x) - x]_{1}^{10}

Compute the boundaries:

10 ln (10) - 9

= \frac{1}{ln ( 10 )} (10 ln (10) - 9)

Simplify

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

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What is the integral from 1 to 10 of ((ln(x))/(ln(10))) ?
The integral from 1 to 10 of ((ln(x))/(ln(10))) is (10ln(10)-9)/(ln(10))