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

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

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

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

1 - \frac{2}{e}

Decimal

0.26424 \dots

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

\int 1 d x = x

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

Compute the boundaries:

- 1 + \frac{2}{e}

= - (- 1 + \frac{2}{e})

Simplify

= 1 - \frac{2}{e}

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