What is the integral from 0 to 1 of xcos(2x) ?

The integral from 0 to 1 of xcos(2x) is 1/4 (2sin(2)+cos(2)-1)

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integral from 0 to 1 of xcos(2x)

\int_{0}^{1} x cos (2 x) d x

\frac{1}{4} (2 sin (2) + cos (2) - 1)

Decimal

0.10061 \dots

Solution steps

\int_{0}^{1} x cos (2 x) d x

Apply u-substitution

: \int_{0}^{2} \frac{u cos ( u )}{4} d u

= \int_{0}^{2} \frac{u cos ( u )}{4} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int_{0}^{2} u cos (u) d u

Apply Integration By Parts

: [u sin (u) - \int sin (u) d u]_{0}^{2}

= \frac{1}{4} [u sin (u) - \int sin (u) d u]_{0}^{2}

\int sin (u) d u = - cos (u)

= \frac{1}{4} [u sin (u) - (- cos (u))]_{0}^{2}

Simplify

= \frac{1}{4} [u sin (u) + cos (u)]_{0}^{2}

Compute the boundaries:

2 sin (2) + cos (2) - 1

= \frac{1}{4} (2 sin (2) + cos (2) - 1)

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What is the integral from 0 to 1 of xcos(2x) ?
The integral from 0 to 1 of xcos(2x) is 1/4 (2sin(2)+cos(2)-1)