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5.4. Homework Problems

1. Graph the function y = ex-7x and get an approximate idea of where any of its zeroes are (i.e., for what values of x we have y(x) = 0). Use Newton's method to find the zeroes to three significant figures of precision.

2. The relationship between x and y is given by xy = sin y + x2y2.
(a) Use Newton's method to find the nonzero solution for y when x=3. Answer: y = 0.2231
(b) Find dy/dx in terms of x and y, and evaluate the derivative at the point on the curve you found in part a. Answer: dy/dx = -0.0379
Based on an example by Craig B. Watkins.

3. Suppose you want to evaluate

\[ \int \frac{dx}{1+\sin 2x} , \]

and you've found

\[ \int \frac{dx}{1+\sin x} = -\tan\left(\frac{\pi}{4}-\frac{x}{2}\right) \]

in a table of integrals. Use a change of variable to find the answer to the original problem.

4. Evaluate

\[ \int \frac{\sin x dx}{1+\cos x} . \]

5. Evaluate

\[ \int \frac{\sin x dx}{1+\cos^2 x} . \]

6. Evaluate

\[ \int x\sqrt{a-x}dx . \]

7. Evaluate

\[ \int \sqrt{x^4+bx^2} dx , \]

where b is a constant.

8. Evaluate

\[ \int x e^{-x^2} dx . \]

9. Evaluate

\[ \int x e^x dx . \]

10. Use integration by parts to evaluate the following integrals.

\begin{gather} & \int \sin^{-1} xdx \\ & \int \cos^{-1} xdx \\ & \int \tan^{-1} xdx \end{gather}

11. Evaluate

\[ \int x^2 \sin x dx . \]

Hint: Use integration by parts more than once.

12. Evaluate

\[ \int \frac{dx}{x^2-x-6} . \]

13. Evaluate

\[ \int \frac{dx}{x^3+3x^2-4} . \]

14. Evaluate

\[ \int \frac{dx}{x^3-x^2+4x-4} . \]

15. Apply integration by parts twice to

\[ \int e^{-x}\cos x dx , \]

examine what happens, and manipulate the result in order to solve the original integral. (An approach that doesn't rely on tricks is given in Example 91 on p. 123.)

16. Plan, but do not actually carry out the steps that would be required in order to generalize the result of Example 70 on p. 91 in order to evaluate

\[ \int x^a b^{-x} dx , \]

where a and b are constants. Which is easier, the generalization from 2 to a, or the one from e to b? Do we need to introduce any restrictions on a or b?

17. The integral \(\int e^{-x^2}dx\) can't be done in closed form. Knowing this, use a change of variable to write down a different integral that also can't be done in closed form.

18. Consider the integral

\[ \int e^{x^p} dx , \]

where p is a constant. There is an obvious substitution. If this is to result in an integral that can be evaluated in closed form by a series of integrations by parts, what are the possible values of p? Don't actually complete the integral; just determine what values of p will work.

19. Evaluate the hundredth derivative of the function (x2+1)/(x3-x) using paper and pencil. [Vladimir Arnol'd]