What is the integral from 1 to e^pi of cos(ln(x)) ?

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

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integral from 1 to e^pi of cos(ln(x))

\int_{1}^{e^{π}} cos (ln (x)) d x

\frac{- e ^{π} - 1}{2}

Decimal

- 12.07034 \dots

Solution steps

\int_{1}^{e^{π}} cos (ln (x)) d x

Compute the indefinite integral:

\int cos (ln (x)) d x = \frac{1}{2} x sin (ln (x)) + \frac{1}{2} x cos (ln (x)) + C

Compute the boundaries:

\int_{1}^{e^{π}} cos (ln (x)) d x = - \frac{e ^{π}}{2} - \frac{1}{2}

= - \frac{e ^{π}}{2} - \frac{1}{2}

Simplify

= \frac{- e ^{π} - 1}{2}

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