$\int_{0}^{\infty} \frac{1}{(x^a+x^b)^p} dx$ convergence

Question

Can anyone help me how to determine if the following integral converges in case of $p$? I don't have any idea what should I do. Thanks $$\int_{0}^{\infty} \frac{1}{(x^a+x^b)^p} dx,$$

where $0<a \leq b < \infty$.

Answer 1

Break your integral in two:$$\int_0^1\frac{\mathrm dx}{(x^a+x^b)^p}+\int_1^\infty\frac{\mathrm dx}{(x^a+x^b)^p}.$$Since$$\lim_{x\to\infty}\frac{\frac1{x^{bp}}}{\frac1{(x^a+x^b)^p}}=1,$$the second integral converges if and only if the integral$$\int_1^\infty\frac{\mathrm dx}{x^{bp}}$$converges. And, since$$\lim_{x\to0}\frac{\frac1{x^{ap}}}{\frac1{(x^a+x^b)^p}}=1,$$the first integral converges if and only if the integral$$\int_0^1\frac{\mathrm dx}{x^{ap}}$$converges. Can you take it from here?