What is \int e^{x}\{1+\ln x+x\ln x\}\,dx equal to?
- A. xe^{x}\ln x+c ✓
- B. x^{2}e^{x}\ln x+c
- C. x+e^{x}\ln x+c
- D. xe^{x}+\ln x+c
Correct Answer: A. xe^{x}\ln x+c
Explanation
The given integral can be rearranged as \int (e^x \ln x + x e^x \ln x + e^x) \,dx. Let's differentiate y = x e^x \ln x. Using the product rule: \frac{dy}{dx} = 1 \cdot e^x \ln x + x \cdot e^x \ln x + x e^x \cdot \frac{1}{x} = e^x \ln x + x e^x \ln x + e^x. This is exactly the integrand. Therefore, the integral is x e^x \ln x + c.
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