Evaluate the following integral:∫0𝜋/2𝑥sin⁡(𝑥) 𝑑𝑥∫ 0π/2​ xsin(x)dx

To evaluate the integral ∫ from 0 to π/2 of x*sin(x) dx, we can use integration by parts. The formula for integration by parts is ∫udv = uv - ∫vdu.

Let's choose: u = x (so du = dx) dv = sin(x) dx (so v = -cos(x))

Now, we substitute into the formula:

∫ from 0 to π/2 of xsin(x) dx = [uv] from 0 to π/2 - ∫ from 0 to π/2 of v du = [x(-cos(x))] from 0 to π/2 + ∫ from 0 to π/2 of cos(x) dx = [π/2 * -cos(π/2) - 0 * -cos(0)] + [sin(x)] from 0 to π/2 = [0 - 0] + [sin(π/2) - sin(0)] = 0 + 1 - 0 = 1

So, ∫ from 0 to π/2 of x*sin(x) dx = 1.