What is the integral from 1/2 to e of ln(x) ?

The integral from 1/2 to e of ln(x) is 1/2 ln(2)+1/2

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integral from 1/2 to e of ln(x)

\int_{\frac{1}{2}}^{e} ln (x) d x

\frac{1}{2} ln (2) + \frac{1}{2}

Decimal

0.84657 \dots

Solution steps

\int_{\frac{1}{2}}^{e} ln (x) d x

Apply Integration By Parts

: [x ln (x) - \int 1 d x]_{\frac{1}{2}}^{e}

= [x ln (x) - \int 1 d x]_{\frac{1}{2}}^{e}

\int 1 d x = x

= [x ln (x) - x]_{\frac{1}{2}}^{e}

Compute the boundaries:

- (- \frac{1}{2} ln (2) - \frac{1}{2})

= - (- \frac{1}{2} ln (2) - \frac{1}{2})

Simplify

= \frac{1}{2} ln (2) + \frac{1}{2}

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What is the integral from 1/2 to e of ln(x) ?
The integral from 1/2 to e of ln(x) is 1/2 ln(2)+1/2