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6.3. Homework Problems

1. 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.
(b) Find the average value of x, or show that it diverges.
(c) Find the standard deviation of x, or show that it diverges.

8. Prove

\[ \int_0^\infty e^{-x}x^ndx = n! . \]