Integrate xln(2x)
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.
To integrate xln(2x), we can use integration by parts. The formula for integration by parts is:
∫u * v dx = u * ∫v dx - ∫(u' * ∫v dx) dx
Let's choose:
u = ln(2x) and dv = x dx
To find du, we differentiate u with respect to x:
du/dx = 1/x
To find v, we integrate dv with respect to x:
∫x dx = (x^2)/2
Now we can substitute these values into the integration by parts formula:
∫xln(2x) dx = ln(2x) * (x^2)/2 - ∫(1/x * (x^2)/2) dx
Simplifying the expression, we have:
∫xln(2x) dx = (1/2) * x^2ln(2x) - (1/2) * ∫x dx
∫xln(2x) dx = (1/2) * x^2ln(2x) - (1/2) * (x^2/2) + C
Therefore, the integral of xln(2x) is (1/2)x^2ln(2x) - (1/4)x^2 + C, where C is the constant of integration.