7.31 Evaluate the following integrals: (a) ∫-∞^∞ e^-4t^2δ(t...

7.31 Evaluate the following integrals: (a) $\int_{-\infty}^{\infty} e^{-4t^2} \delta(t - 2)dt$ (b) $\int_{-\infty}^{\infty} [5\delta(t) + e^{-t}\delta(t) + \cos 2\pi t \delta(t)]dt$

Transcript

00:01 Hello students, in this question we have to evaluate the following integrals.

00:04 Here the first integral is 5x square plus 1 by x power 4 plus 2 into dx.

00:13 Here we can rewrite this as 5 into integral x square dx plus integral 1 by x power 4 into dx plus 2 into integral dx.

00:26 By integrating x square we will get x cube by 3 plus by integrating 1 by x power 4 we will get minus 1 by 3 x cube plus 2 into x plus c.

00:41 Therefore, the integral 5 x square plus 1 by x power 4 plus 2 into dx is 5 by 3 x cube minus 1 by 3 x cube plus 2 x plus c.

01:00 Next we have to evaluate the integral cos 3x plus e power 4x into dx.

01:11 Here we can write this as integral cos 3x into dx plus integral e power 4x into dx.

01:21 By integrating cos dx we will get sin 3x by 3 plus by integrating e power 4x we will get 1 by 4 into e power 4x plus integration constant c.

01:35 Therefore, integral of cos 3x plus e power 4x into dx equal to sin 3x by 3 plus e power 4x by 4 plus.

01:55 Next we have to evaluate the integral cos of lon x by x into dx.

02:05 Here we have to consider u equal to lon x.

02:12 Therefore, du becomes 1 by x into dx.

02:16 Therefore, the integration becomes integral of cos of lon x by x into dx equal to integral cos u into du.

02:31 By integrating cos u we will get sin u plus constant c.

02:36 Here we have to replace u by lon x.

02:39 Therefore, it will be sin of lon x plus c...

Question

7.31 Evaluate the following integrals: (a) $\int_{-\infty}^{\infty} e^{-4t^2} \delta(t - 2)dt$ (b) $\int_{-\infty}^{\infty} [5\delta(t) + e^{-t}\delta(t) + \cos 2\pi t \delta(t)]dt$

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