Integrate ln[(x)(1-x)] in the limits 0 to 1

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Rewrite Integral: Rewrite the integral to make it easier to work with. ∫ ln(x(1-x)) dx from 0 to 1Split Logarithm: Use the property of logarithms to split the ln function. ∫ (ln(x) + ln(1-x)) dx from 0 to 1Split Integrals: Split the integral into two separate integrals. ∫ ln(x) dx + ∫ ln(1-x) dx from 0 to 1Integration by Parts: Use integration by parts for ∫ ln(x) dx where u=ln(x) and dv=dx. Let u = ln(x), then du = (1/x) dx. Let dv = dx, then v = x. Now, integrate by parts: ∫ u dv = uv - ∫ v du.Apply Integration by Parts: Apply integration by parts to the first integral. x ln(x) - ∫ x * (1/x) dx from 0 to 1 x ln(x) - ∫ 1 dx from 0 to 1Simplify and Integrate: Simplify and integrate. x ln(x) - x from 0 to 1Integration by Parts (ln(1-x)): Now, use integration by parts for ∫ ln(1-x) dx where u=ln(1-x) and dv=dx. Let u = ln(1-x), then du = -(1/(1-x)) dx. Let dv = dx, then v = x. Now, integrate by parts: ∫ u dv = uv - ∫ v du.Apply Integration by Parts: Apply integration by parts to the second integral. x ln(1-x) + ∫ x * (1/(1-x)) dx from 0 to 1 x ln(1-x) + ∫ x / (1-x) dx from 0 to 1Complex Integral (ln(1-x) dx): The integral ∫ x / (1-x) dx is more complex and requires partial fractions or another method to solve. However, we can notice that the function ln(1-x) is not defined at x=1, which means the integral does not converge.

Integrate $\ln[(x)(1-x)]$ in the limits $0$ to $1$

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