6.3. Homework Problems1. Integrate \[ \int_0^\infty e^{x} dx , \] or show that it diverges. 2. Integrate \[ \int_1^\infty \frac{dx}{x} , \] or show that it diverges. 3. Integrate \[ \int_0^1 \frac{dx}{x} , \] or show that it diverges. 4. Integrate \[ \int_0^\infty x^2 2^{x} dx , \] or show that it diverges. (solution in the pdf version of the book) 5. Integrate \[ \int_0^\infty e^{x}\cos x dx \] or show that it diverges. (Problem 15 on p. 99 suggests a trick for doing the indefinite integral.) 6. Prove that \[ \int_0^\infty e^{e^x} dx \] converges, but don't evaluate it. 7.
(a) Verify that the probability distribution dP/dx given in Example 60 on page 80
is properly normalized. 8. Prove \[ \int_0^\infty e^{x}x^ndx = n! . \]
