Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §7.1 Integration by Parts, #48 · Calculus · Techniques of integration

(c)

44–54 are about optional topics at the end of the section.

In $δ (x - \frac{1}{2})$ , the spike shifts to $x = \frac{1}{2}$ . It is the derivative of the shifted step $U (x - \frac{1}{2})$ . The integral of $v (x) δ (x - \frac{1}{2})$ equals the value of $v$ at $x = \frac{1}{2}$ . Compute

(a) $\int_{0}^{1} δ (x - \frac{1}{2}) d x$ ;

(b) $\int_{0}^{1} e^{x} δ (x - \frac{1}{2}) d x$ ;

(c) $\int_{- 1}^{1} δ (x) δ (x - \frac{1}{2}) d x$ .

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §7.1 Integration by Parts, #48

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