Question

First make a substitution and then use integration by parts to evaluate the integral. (Use c for the constant of integration.) ∫(x ln(1 - x)) dx

Answer 1

We solve the integral ∫(x ln(1 - x)) dx by substitution and integration by parts, starting with u = ln(1 - x) and applying the integration by parts formula to find the solution with the constant of integration c.

Step-by-step explanation:

To evaluate the integral ∫(x ln(1 - x)) dx, we start with a substitution. Let u = ln(1 - x), which implies du = -⅓ dx. We can then rewrite the integral in terms of u and apply integration by parts, which states that ∫u dv = uv - ∫v du. We choose dv to be the remaining part of the integrand after substitution. After finding the antiderivatives and performing the necessary algebraic manipulations, we can substitute back the original variable x to obtain the result with the constant of integration c.