Let a function h(x) be defined as h(x) = 0, for all x 0. Also _ -… - Vidyadip

MCQ

Let a function $h(x)$ be defined as $h(x) = 0$, for all $x \ne 0$. Also $\int\limits_{ - \,\infty }^\infty {{\rm{h}}(x)\,\cdot\,{\rm{f}}(x)\,dx} $ $= f (0)$, for every function $f (x)$. Then the value of the definite integral $\int\limits_{ - \,\infty }^\infty {{\rm{h}}'(x)\,\cdot\,\sin x\,dx} $, is

A
equal to zero
B
equal to $1$
✓
equal to $- 1$
D
non existent

✓

Answer

Correct option: C.

equal to $- 1$

c
$I =$ $\int\limits_{ - \,\infty }^\infty {\underbrace {{\rm{h}}'(x)}_{II}\,\cdot\,\underbrace {\sin x}_I\,dx} $ $= \sin x\,\cdot\,\left. {h(x)} \right]_{ - \,\infty }^{\,\infty }$ $- \int\limits_{ - \,\infty }^\infty {\cos x\,\cdot\,h(x)\,dx} $ $= 0 - \cos \,0 = - 1$
note that here $\cos\, x = f (x)$

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