The integral ∫(xˣ)(1 + ln x)dx is of which form?

The integral ∫(xˣ)(1 + ln x)dx is of which form?

• A. ∫eˣ[f(x)+f'(x)]dx
• B. Substitution u = xˣ
• C. Integration by parts
• D. Standard formula

Answer: A) ∫eˣ[f(x)+f'(x)]dx

Explanation: xˣ = e^(x ln x). d/dx(x ln x) = 1 + ln x. So the integral is ∫e^(x ln x) d/dx(x ln x) dx = ∫eᵘ du = eᵘ + C = xˣ + C. This is the form ∫eˣ[f+f'].