Question

求不定积分x^2e^x/(2+x)^2dx

Answer 1

∫ x²e^x/(2 + x)² dx

= - ∫ x²e^x d[1/(2 + x)]

= - x²e^x/(2 + x) + ∫ 1/(2 + x) d(x²e^x)

= - x²e^x/(2 + x) + ∫ 1/(2 + x) * (2 + x)xe^x dx

= - x²e^x/(2 + x) + ∫ xe^x dx

= - x²e^x/(2 + x) + ∫ x de^x

= - x²e^x/(2 + x) + xe^x - ∫ e^x dx

= - x²e^x/(2 + x) + xe^x - e^x + C

= [- x²e^x + (2 + x)xe^x - (2 + x)e^x]/(2 + x) + C

= (-x²e^x + x²e^x + 2xe^x - 2e^x - xe^x)/(2 + x) + C

= (xe^x - 2e^x)/(2 + x) + C

= [(x - 2)/(x + 2)]e^x + C