91Ó°ÊÓ

Show that ${\mathbf{∫}}_{\mathbf{a}}^{\mathbf{b}}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{k}\mathit{x}\mathit{d}\mathit{x}\mathbf{=}{\mathbf{∫}}_{\mathbf{a}}^{\mathbf{b}}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathit{k}\mathit{x}\mathit{d}\mathit{x}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{(}\mathit{b}\mathbf{-}\mathit{a}\mathbf{)}$ if$\mathit{k}\mathbf{(}\mathit{b}\mathbf{}\mathbf{-}\mathbf{}\mathit{a}\mathbf{)}$is an integral multiple of$\mathit{π}$, or if kb and ka are both integral multiples of $\frac{\mathbf{π}}{\mathbf{2}}$.

Given algebraic proofs forodd and even functions:

1. even times = even; odd times odd = even; even times odd = odd;
2. the derivative of an even function is odd; the derivative of an odd function is even.

Use the results ${\mathbf{∫}}_{\mathbf{a}}^{\mathbf{b}}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{k}\mathit{x}\mathit{d}\mathit{x}\mathbf{=}{\mathbf{∫}}_{\mathbf{a}}^{\mathbf{b}}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathit{k}\mathit{x}\mathit{d}\mathit{x}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{(}\mathit{b}\mathbf{-}\mathit{a}\mathbf{)}$ to evaluate the following integrals without calculation.

(a)${\mathbf{∫}}_{\mathbf{0}}^{\mathbf{4}\mathbf{π}\mathbf{/}\mathbf{3}}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\left(\frac{3x}{2}\right)\mathit{d}\mathit{x}$

(b)${\mathbf{∫}}_{\mathbf{-}\mathbf{π}\mathbf{/}\mathbf{2}}^{\mathbf{3}\mathbf{π}\mathbf{/}\mathbf{2}}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\left(\frac{x}{2}\right)\mathit{d}\mathit{x}$

In Problems 13to 16, find the Fourier cosine transform of the function in the indicated problem, and write f(x)as the Fourier integral [ use equation (12.15)]. Verify that the cosine integral for f(x)is the same as the exponential integral found previously.

14. Problem 7

Use Problem 5.7to show that$\frac{\mathbf{1}}{{\mathbf{2}}^{\mathbf{2}}\mathbf{-}\mathbf{1}}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{4}}^{\mathbf{2}}\mathbf{-}\mathbf{1}}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{6}}^{\mathbf{2}}\mathbf{-}\mathbf{1}}\mathbf{+}\mathbf{⋯}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}$

In Problems 13to 16, find the Fourier cosine transform of the function in the indicated problem, and write f(x)as the Fourier integral [ use equation (12.15)]. Verify that the cosine integral for f(x)is the same as the exponential integral found previously.

15. Problem 9

Givenf(x) = x for 0 < x < 1, sketch the even function f_c of period 2 and the odd functions f_s of period 2, each of which equals f(x) on 0 < x < 1. Expand f_c in a cosine series and f_sin a sine series.

Use the results ${\mathbf{∫}}_{\mathbf{a}}^{\mathbf{b}}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{k}\mathit{x}\mathit{d}\mathit{x}\mathbf{=}{\mathbf{∫}}_{\mathbf{a}}^{\mathbf{b}}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathit{k}\mathit{x}\mathit{d}\mathit{x}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{(}\mathit{b}\mathbf{-}\mathit{a}\mathbf{)}\mathit{a}$ to evaluate the following integrals without calculation.

(a)${\mathbf{∫}}_{\mathbf{-}\mathbf{1}\mathbf{/}\mathbf{4}}^{\mathbf{11}\mathbf{/}\mathbf{4}}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathit{π}\mathit{x}\mathit{d}\mathit{x}$

(b)${\mathbf{∫}}_{\mathbf{-}\mathbf{1}}^{\mathbf{2}}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\left(\frac{\mathrm{π}x}{3}\right)\mathit{d}\mathit{x}$

Use the results${\mathbf{∫}}_{\mathbf{a}}^{\mathbf{b}}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{k}\mathit{x}\mathit{d}\mathit{x}\mathbf{=}{\mathbf{∫}}_{\mathbf{a}}^{\mathbf{b}}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathit{k}\mathit{x}\mathit{d}\mathit{x}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{(}\mathit{b}\mathbf{-}\mathit{a}\mathbf{)}$ to evaluate the following integrals without calculation.

(a)${\mathbf{∫}}_{\mathbf{0}}^{\mathbf{2}\mathbf{Ï€}\mathbf{/}\mathbf{Ó\neg }}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{Ó\neg }\mathit{t}\mathit{d}\mathit{t}$

(b)${\mathbf{∫}}_{\mathbf{0}}^{\mathbf{2}}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathbf{2}\mathit{π}\mathit{t}\mathit{d}\mathit{t}$

In Problems 13to 16, find the Fourier cosine transform of the function in the indicated problem, and write f(x)as the Fourier integral [ use equation (12.15)]. Verify that the cosine integral forf(x) is the same as the exponential integral found previously.

16. Problem 11