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

The integral from 0 to pi of xcos(2x) is 0

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integral from 0 to pi of xcos(2x)

\int_{0}^{π} x cos (2 x) d x

0

Solution steps

\int_{0}^{π} x cos (2 x) d x

Apply u-substitution

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

= \int_{0}^{2 π} \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 π} u cos (u) d u

Apply Integration By Parts

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

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

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

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

Simplify

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

Compute the boundaries:

0

= \frac{1}{4} \cdot 0

Simplify

= 0

\frac{32}{( s ^{2} + 4 ^{2} ) ( s + 4 ) ^{2}}

x \to 0 + lim (ln (\frac{4}{x ^{3}}))

x \to 3 lim (8 x + 2)

x \to 5 lim (∣ x ∣)

\int \frac{x + 25}{x} d x

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