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

The integral from 0 to pi of x*sin(2x) is -pi/2

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integral from 0 to pi of x*sin(2x)

\int_{0}^{π} x \cdot sin (2 x) d x

- \frac{π}{2}

Decimal Notation

- 1.57079 \dots

Solution steps

\int_{0}^{π} x sin (2 x) d x

Apply Integration By Parts

: [\frac{1}{2} (- x cos (2 x) - 2 \cdot \int - \frac{1}{2} cos (2 x) d x)]_{0}^{π}

= [\frac{1}{2} (- x cos (2 x) - 2 \cdot \int - \frac{1}{2} cos (2 x) d x)]_{0}^{π}

\int - \frac{1}{2} cos (2 x) d x = - \frac{1}{4} sin (2 x)

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

Simplify

[\frac{1}{2} (- x cos (2 x) - 2 (- \frac{1}{4} sin (2 x)))]_{0}^{π} : [\frac{1}{2} (- x cos (2 x) + \frac{1}{2} sin (2 x))]_{0}^{π}

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

Compute the boundaries:

- \frac{π}{2}

= - \frac{π}{2}

\frac{d y}{d x} = (x - 2) e^{- 2 y}, y (2) = ln (2)

\int_{0}^{\frac{π}{3}} sin (3 x) d x

\int_{- 4}^{4} 90 d t

x \to 0 lim (\frac{5}{x ^{8}})

\frac{\partial}{\partial x} (e^{3 x} cos (6 y))

What is the integral from 0 to pi of x*sin(2x) ?
The integral from 0 to pi of x*sin(2x) is -pi/2