Question

当n为多少 ∫2~+∞ x /√[（x^2 +3）^n] dx 收敛

Answer 1

∫x/√[(x²+3)^n]dx (2→+∞)

=½∫1/[(x²+3)^(n/2)]dx² (2→+∞)

=½∫[(x²+3)^(-n/2)]d(x²+3) (2→+∞)

=[1/(2-n)](x²+3)/(x²+3)^(n/2)] (2→+∞)

当 n>2,(x²+3)^(n/2)>(x²+3),结果收敛.