∫π0xsin3xdx

To solve the integral ∫ from 0 to π of x*sin(3x) dx, we can use integration by parts. The formula for integration by parts is ∫udv = uv - ∫vdu.

Step 1: Identify u and dv Let's choose u = x and dv = sin(3x) dx.

Step 2: Compute du and v Then, du = dx and v = -1/3 cos(3x).

Step 3: Apply the integration by parts formula ∫x sin(3x) dx = uv - ∫v du = x * (-1/3 cos(3x)) - ∫(-1/3 cos(3x)) dx = -x/3 cos(3x) + 1/3 ∫cos(3x) dx

Step 4: Solve the remaining integral The integral of cos(3x) dx is 1/3 sin(3x). So, = -x/3 cos(3x) + 1/3 * 1/3 sin(3x) = -x/3 cos(3x) + 1/9 sin(3x)

Step 5: Evaluate from 0 to π = [-π/3 cos(3π) + 1/9 sin(3π)] - [-0/3 cos(0) + 1/9 sin(0)] = [π/3 + 0] - [0 + 0] = π/3

So, ∫ from 0 to π of x*sin(3x) dx = π/3.