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

The integral from 1 to e of sin(x)ln(x) is \Ci(e)-0.03434…

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

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

Ci (e) - 0.03434 \dots

Solution steps

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

Apply Integration By Parts

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

\int - \frac{cos ( x )}{x} d x = - Ci (x)

= [- ln (x) cos (x) - (- Ci (x))]_{1}^{e}

Simplify

= [- ln (x) cos (x) + Ci (x)]_{1}^{e}

Compute the boundaries:

- cos (e) + Ci (e) - 0.9460830704

= - cos (e) + Ci (e) - 0.94608 \dots

Simplify

= Ci (e) - 0.03434 \dots

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What is the integral from 1 to e of sin(x)ln(x) ?
The integral from 1 to e of sin(x)ln(x) is \Ci(e)-0.03434…