Evaluate ∫[(1-x)√x] dx

To evaluate the integral ∫[(1-x)√x] dx, we can use the method of integration by parts. The formula for integration by parts is ∫udv = uv - ∫vdu.

Step 1: Identify u and dv Let's choose u = 1-x and dv = √x dx.

Step 2: Compute du and v The derivative of u (du) is -dx. To find v, we integrate dv, which gives us v = (2/3)x^(3/2).

Step 3: Apply the integration by parts formula Now we substitute u, v, du, and dv into the integration by parts formula: ∫udv = uv - ∫vdu = (1-x)(2/3)x^(3/2) - ∫(2/3)x^(3/2)(-dx) = (2/3)x^(3/2) - x + ∫(2/3)x^(3/2) dx

Step 4: Simplify the integral The integral ∫(2/3)x^(3/2) dx can be computed directly: = (2/3) * (2/5)x^(5/2) = (4/15)x^(5/2)

Step 5: Combine all terms So the original integral ∫[(1-x)√x] dx becomes: = (2/3)x^(3/2) - x + (4/15)x^(5/2) + C where C is the constant of integration.