Integrate xln(2x)

Answer 1

To integrate xln(2x), we can use integration by parts. The formula for integration by parts is:

∫ u dv = uv - ∫ v du

Let's choose u = ln(2x) and dv = x dx.

Differentiating u, we have du/dx = 1/(2x) * 2 = 1/x.

To find v, we integrate dv: ∫ x dx = (1/2)x^2.

Now we can use the formula:

∫ xln(2x) dx = uv - ∫ v du

Substituting the values, we get:

= ln(2x) * (1/2)x^2 - ∫ (1/2)x^2 * (1/x) dx

= (1/2) ln(2x) * x^2 - (1/2) ∫ x dx

= (1/2) ln(2x) * x^2 - (1/2) * (1/2) x^2 + C

= (1/2) ln(2x) * x^2 - (1/4) x^2 + C

Therefore, the integral of xln(2x) is (1/2) ln(2x) * x^2 - (1/4) x^2 + C, where C is the constant of integration.