Evaluate the following integrals:

(a)${\mathbf{\int }}_{\mathbf{-}\mathbf{3}}^{\mathbf{+}\mathbf{1}}\mathbf{(}{\mathit{x}}^{\mathbf{3}}\mathbf{-}\mathbf{3}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathit{x}\mathbf{-}\mathbf{1}\mathbf{)}\mathit{\delta }\mathbf{(}\mathit{x}\mathbf{+}\mathbf{2}\mathbf{)}\mathit{d}\mathit{x}$.

(b).${\mathbf{\int }}_{\mathbf{0}}^{\mathbf{\infty }}\mathbf{[}\mathbf{cos}\left(3x\right)\mathbf{+}\mathbf{2}\mathbf{]}\mathbf{\delta }\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{\pi }\mathbf{)}\mathbf{dx}$

(c)${\mathbf{\int }}_{\mathbf{\_}\mathbf{1}}^{\mathbf{+}\mathbf{1}}\mathbf{exp}\mathbf{(}\mathbf{lxl}\mathbf{+}\mathbf{3}\mathbf{)}\mathit{\delta }\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{2}\mathbf{)}\mathit{d}\mathit{x}$

In mathematics, an integral lends numerical values to functions to represent concepts like volume, area, and displacement that result from combining infinitesimally small amounts of data. In mathematics, an integral lends numerical values to functions to represent concepts like volume, area, and displacement that result from combining infinitesimally small amounts of data.

Let,

${\int }_{-3}^{+1}\left(x^3-3x^2+2x-1\right)\delta \left(x+2\right)dx$

According to impulse property,

$$\begin{gathered} x+2=0 \\ x=-2 \end{gathered}$$

In given function, substitute$x=-2$.

$$\begin{gathered} \left(x^3-3x^2+2x-1\right)={(-2)}^3-3{(-2)}^2+2(-2)-1 \\ =-8-12-4-1 \\ =-25 \end{gathered}$$

Hence the value of is ${\int }_{-3}^{+1}(x^3-3x^2+2x-1)\delta (x+2)dx=-25$.

Let,

${\int }_0^{\infty }\left[\cos \left(3x\right)+2\right]\delta \left(x-\mathrm{\pi }\right)dx$

In given function, substitute $x=\mathrm{\pi }$, $\left(\begin{array}{c}x-\mathrm{\pi }=0\\ x=\mathrm{\pi }\\ \end{array}\right)$.

Then,

$$\begin{gathered} \cos \left(3\mathrm{\pi }\right)+2=-1+2 \\ =1 \end{gathered}$$

Hence the value of ${\int }_0^{\infty }\left[\cos \left(3x\right)+2\right]\delta \left(x-\mathrm{\pi }\right)dx$is, 1.

Let,

${\int }_{-1}^{+}\exp (\mathrm{l}\times \mathrm{l}+3)\delta (x-2)dx$

Then,

${\int }_{-1}^{+1}\exp \left(\mathrm{l}\times \mathrm{l}+3\right)\delta \left(x-2\right)dx=0$

0 ($x=2$is outside the domain of integration).

Because x=2 is out of the integration interval.