Question

Use 21 (A), (b), and (c) And use 11.19 21. Make use of the operator equations (11.19) and previous equations to evaluate the following integrals. (a) \int_0^3 (5x - 2)\delta(2 - x) dx (c) \int_{-1}^1 \cos x \delta(-2x) dx (b) \int_0^\infty \phi(x)\delta(x^2 - a^2) dx (d) \int_{-\pi/2}^{\pi/2} \cos x \delta(\sin x) dx We list a few more operator equations (see Problems 20 and 21). (a) \delta(-x) = \delta(x) and \delta(x - a) = \delta(a - x); (b) \delta'(-x) = -\delta'(x) and \delta'(x - a) = -\delta'(a - x); (11.19) (c) \delta(\alpha x) = \frac{1}{|\alpha|}\delta(x), \alpha \neq 0; (d) \delta[(x - a)(x - b)] = \frac{1}{|a - b|}[\delta(x - a) + \delta(x - b)], a \neq b; (e) \delta[f(x)] = \sum_i \frac{\delta(x - x_i)}{|f'(x_i)|}, if f(x_i) = 0 and f'(x_i) \neq 0.

          Use 21 (A), (b), and (c)
And use 11.19
21. Make use of the operator equations (11.19) and previous equations to evaluate the
following integrals.
(a) \int_0^3 (5x - 2)\delta(2 - x) dx
(c) \int_{-1}^1 \cos x \delta(-2x) dx
(b) \int_0^\infty \phi(x)\delta(x^2 - a^2) dx
(d) \int_{-\pi/2}^{\pi/2} \cos x \delta(\sin x) dx
We list a few more operator equations (see Problems 20 and 21).
(a) \delta(-x) = \delta(x) and \delta(x - a) = \delta(a - x);
(b) \delta'(-x) = -\delta'(x) and \delta'(x - a) = -\delta'(a - x);
(11.19)
(c) \delta(\alpha x) = \frac{1}{|\alpha|}\delta(x), \alpha \neq 0;
(d) \delta[(x - a)(x - b)] = \frac{1}{|a - b|}[\delta(x - a) + \delta(x - b)], a \neq b;
(e) \delta[f(x)] = \sum_i \frac{\delta(x - x_i)}{|f'(x_i)|}, if f(x_i) = 0 and f'(x_i) \neq 0.

Use 21 (A), (b), and (c)
And use 11.19
21. Make use of the operator equations (11.19) and previous equations to evaluate the
following integrals.
(a) ∫0^3 (5x - 2)δ(2 - x) dx
(c) ∫-1^1 cosx δ(-2x) dx
(b) ∫0^∞ϕ(x)δ(x^2 - a^2) dx
(d) ∫-π/2^π/2 cosx δ(sinx) dx
We list a few more operator equations (see Problems 20 and 21).
(a) δ(-x) = δ(x) and δ(x - a) = δ(a - x);
(b) δ'(-x) = -δ'(x) and δ'(x - a) = -δ'(a - x);
(11.19)
(c) δ(αx) = (1)/(|α|)δ(x), α≠0;
(d) δ[(x - a)(x - b)] = (1)/(|a - b|)[δ(x - a) + δ(x - b)], a ≠b;
(e) δ[f(x)] = (δ(x - xi))/(|f'(xi)|), if f(xi) = 0 and f'(xi) ≠0.

Added by Mark J.

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