Final answer:
To evaluate the integral from 1 to 3 of (ln(x))² times x³, integration by parts is needed, systematically applying u = (ln(x))² and dv = x³ dx, then differentiating and integrating these selections. After completing the integration process, we evaluate the antiderivative from 1 to 3 to find the answer.
Step-by-step explanation:
To evaluate the integral ∫(3 to 1) (ln(x))² * x³ dx, we can approach it by using integration by parts, which is expressed as ∫ u dv = uv - ∫ v du. This technique is useful when dealing with the product of two functions where one is easier to differentiate and the other is easier to integrate.
Let's choose u = (ln(x))² and dv = x³ dx. We then differentiate u to get du = 2ln(x)x⁻¹ dx and integrate dv to get v = ⅔ x⁴. Applying the integration by parts formula gives us:
uv - ∫ v du = ((ln(x))²) (⅔ x⁴) - ∫ (⅔ x⁴) (2ln(x)x⁻¹) dx
This process may need to be repeated, as the resulting integral could still be complex. For this particular integral, a power reduction formula or further applications of integration by parts may be necessary. Be mindful that the steps laid out here are the setup for a potentially longer solution process.
Remember that completing the question requires evaluating the obtained expression from x = 1 to x = 3 after finding the antiderivative.