91Ó°ÊÓ

The theory of diffraction produces the sine integral function \(\operatorname{Si}(x)=\int_{0}^{x} \frac{\sin t}{t} d t .\) Use the Midpoint Rule to approximate \(\left.\mathrm{Si}(1) \text { and } \mathrm{Si}(10) . \text { (Recall that } \lim _{x \rightarrow 0}(\sin x) / x=1 .\right)\) Experiment with the number of subintervals until you obtain approximations that have an error less than \(10^{-3}\). A rule of thumb is that if two successive approximations differ by less than \(10^{-3}\), then the error is usually less than \(10^{-3}\).

Use the approaches discussed in this section to evaluate the following integrals. $$\int_{0}^{\pi / 4} 3 \sqrt{1+\sin 2 x} d x$$

$$\text {Evaluate the following integrals.}$$ $$\int_{0}^{\sqrt{\pi / 2}} x \sin ^{3}\left(x^{2}\right) d x$$

Use a computer algebra system to evaluate the following indefinite integrals. Assume that a is a positive real number. $$\int\left(a^{2}-x^{2}\right)^{-2} d x$$

Evaluate the following integrals or state that they diverge. $$\int_{0}^{\pi / 2} \sec \theta d \theta$$

Use the approaches discussed in this section to evaluate the following integrals. $$\int \frac{e^{x}}{e^{2 x}+2 e^{x}+1} d x$$

The heights of U.S. men are normally distributed with a mean of 69 inches and a standard deviation of 3 inches. This means that the fraction of men with a height between \(a\) and \(b\) (with \(a

Evaluate the following definite integrals. $$\int_{0}^{1 / \sqrt{3}} \sqrt{x^{2}+1} d x$$

Determine whether the following statements are true and give an explanation or counterexample. a. To evaluate \(\int \frac{4 x^{6}}{x^{4}+3 x^{2}} d x\), the first step is to find the partial fraction decomposition of the integrand. b. The easiest way to evaluate \(\int \frac{6 x+1}{3 x^{2}+x} d x\) is with a partial fraction decomposition of the integrand. c. The rational function \(f(x)=\frac{1}{x^{2}-13 x+42}\) has an irreducible quadratic denominator. d. The rational function \(f(x)=\frac{1}{x^{2}-13 x+43}\) has an irreducible quadratic denominator.

Use a computer algebra system to evaluate the following indefinite integrals. Assume that a is a positive real number. $$\int \frac{\left(x^{2}-a^{2}\right)^{3 / 2}}{x} d x$$