Chapter 8.11, Problem 15P

Use (11.6) and (11.14) to (11.16) to evaluate the following integrals.Warning hint:

See comments just after ( 11.6 ) and ( 11.16 ) about the range of integration.

(a) ${\displaystyle {\int }_0^{\pi }\sin }x\delta \left(x-\frac{\pi }{2}\right)dx$ (b) ${\displaystyle {\int }_0^{\pi }\sin }x\delta \left(x+\frac{\pi }{2}\right)dx$

(c) ${\displaystyle {\int }_{-1}^1{e}^{3x}}{\delta }^{\prime }(x)dx$ (d) ${\displaystyle {\int }_0^{\pi }\cosh }x{\delta }^{\prime \prime }(x-1)dx$