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

The integral from 1 to e of cos(ln(x)) is (esin(1)+ecos(1)-1)/2

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

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

\frac{e sin ( 1 ) + e cos ( 1 ) - 1}{2}

Decimal

1.37802 \dots

Solution steps

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

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

Simplify

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

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