Evaluate the integral. int-x ln(-2x)dx Answer:

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Evaluate the integral.  int-x ln(-2x)dx Answer:

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Identify integral: Identify the integral to be solved. We need to evaluate the integral of the function -x ln(-2x) with respect to x. I = ∫-x ln(-2x) dxUse integration by parts: Use integration by parts. Integration by parts formula is ∫u dv = uv - ∫v du, where u and dv are parts of the integrand. Let u = ln(-2x) and dv = -x dx. Then we need to find du and v.Differentiate and integrate: Differentiate u and integrate dv. Differentiating u with respect to x gives us du = (1/x) dx. Integrating dv with respect to x gives us v = -x^2/2.Apply integration by parts: Apply the integration by parts formula. Now we can apply the integration by parts formula: I = uv - ∫v du I = ln(-2x) * (-x^2/2) - ∫(-x^2/2) * (1/x) dxSimplify integral: Simplify the integral. Simplify the integral by canceling x in the second term: I = -x^2/2 * ln(-2x) - ∫(-x/2) dxIntegrate remaining term: Integrate the remaining term. The integral of -x/2 with respect to x is -x^2/4. I = -x^2/2 * ln(-2x) + x^2/4 + C, where C is the constant of integration.Combine and write final answer: Combine the terms and write the final answer. The final answer is: I = -x^2/2 * ln(-2x) + x^2/4 + C

Evaluate the integral. $\newline$ $\int-x \ln (-2 x) d x$ $\newline$ Answer:

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Evaluate the integral. $\newline$ $\int-x \ln (-2 x) d x$ $\newline$ Answer: