Integrate by parts§ln(4x)x^3/3dx? ?

To integrate by parts, we use the formula:

∫u dv = uv - ∫v du

Let's choose:

u = ln(4x) => du = (1/x) dx
dv = x^3/3 dx => v = x^4/12

Now we can apply the formula:

∫ln(4x) * x^3/3 dx = (ln(4x) * x^4/12) - ∫(x^4/12) * (1/x) dx
= (x^4/12)ln(4x) - ∫x^3/12 dx
= (x^4/12)ln(4x) - (1/12)∫x^3 dx
= (x^4/12)ln(4x) - (1/12)(x^4/4) + C
= (x^4/12)ln(4x) - x^4/48 + C

Therefore, the integral of ln(4x) * x^3/3 dx is (x^4/12)ln(4x) - x^4/48 + C.

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