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Which of the following statements are true? (i) The integral \int_0^\infty x...

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Which of the following statements are true?

(i) The integral $\int_0^\infty x^{-3/2} e^{-x} \, dx$ \int_0^\infty x^{-3/2} e^{-x} \, dx converges, whereas the integral $\int_0^\infty x^{-1} e^{-x} \, dx$ \int_0^\infty x^{-1} e^{-x} \, dx diverges.

(ii) The integral $\int_0^1 x^{-3/2}(1 - x)^{-1/2} \, dx$ \int_0^1 x^{-3/2}(1 - x)^{-1/2} \, dx converges, whereas the integral $\int_0^1 x^{-1}(1 - x)^{-1}\, dx$ \int_0^1 x^{-1}(1 - x)^{-1}\, dx diverges.

(iii) The integral $\int_0^\infty x^{-1/2} e^{-x} \, dx$ \int_0^\infty x^{-1/2} e^{-x} \, dx converges, whereas the integral $\int_0^\infty x^{-1} e^{-x} \, dx$ \int_0^\infty x^{-1} e^{-x} \, dx diverges.

(iv) The integral $\int_0^1 x^{-1/2}(1 - x)^{-1/2} \, dx$ \int_0^1 x^{-1/2}(1 - x)^{-1/2} \, dx converges, whereas the integral $\int_0^1 x^{-1}(1 - x)^{-1}\, dx$ \int_0^1 x^{-1}(1 - x)^{-1}\, dx diverges.