Question

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

Short answer

• ${\int }_{-3}^{+1}(x^3-3x^2+2x-1)\delta (x+2)dx=-25$
  
• ${\int }_0^{\infty }[\cos \left(3\mathrm{x}\right)+2]\mathrm{\delta }(\mathrm{x}-\mathrm{\pi })\mathrm{dx}=1$
  
• ${\int }_{\mathrm{\_}1}^{+1}\exp (\mathrm{lxl}+3)\delta (\mathrm{x}-2)dx=0$

Step-by-step solution

Significance of the integral

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.

(a) Evaluating the integral

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

:(b) Evaluating the integral

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.

(c) Evaluating the integral

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.