What is the integral from 0.5 to 1 of cos(pi)xln(x) ?

The integral from 0.5 to 1 of cos(pi)xln(x) is 0.10085…

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integral from 0.5 to 1 of cos(pi)xln(x)

\int_{0.5}^{1} cos (π) x ln (x) d x

0.10085 \dots

Solution steps

\int_{0.5}^{1} cos (π) x ln (x) d x

Take the constant out:

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

= cos (π) \cdot \int_{0.5}^{1} x ln (x) d x

cos (π) = - 1

= (- 1) \cdot \int_{0.5}^{1} x ln (x) d x

Apply Integration By Parts

: [0.5 x^{2} ln (x) - \int 0.5 x d x]_{0.5}^{1}

= (- 1) [0.5 x^{2} ln (x) - \int 0.5 x d x]_{0.5}^{1}

\int 0.5 x d x = 0.25 x^{2}

= (- 1) [0.5 x^{2} ln (x) - 0.25 x^{2}]_{0.5}^{1}

Simplify

= - [0.5 x^{2} ln (x) - 0.25 x^{2}]_{0.5}^{1}

Compute the boundaries:

- 0.10085 \dots

= - (- 0.10085 \dots)

Simplify

= 0.10085 \dots

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What is the integral from 0.5 to 1 of cos(pi)xln(x) ?
The integral from 0.5 to 1 of cos(pi)xln(x) is 0.10085…