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# Answer to Question #243755 in Chemical Engineering for Lokika

Question #243755

∫0^{π}cos3t 𝛿(t-2π) dt. A)-1. B)0. C)1. D) infinity

1
2021-10-06T00:31:10-0400

SOLUTIONS FOR HOMEWORK SECTION 6.4 AND 6.5

Problem 1: For each of the following functions do the following: (i) Write the function as

a piecewise function and sketch its graph, (ii) Write the function as a combination of terms

of the form ua(t)k(t −a) and compute the Laplace transform

(a) f(t) = t(1 −u1(t)) + et(u1(t) −u2(t))

(b) h(t) = sin(2t) + uπ(t)(t/π −sin(2t)) + u2π(t)(2π −t)/π

(c) g(t) = u0(t) + ∑5

k=1(−1)kuk(t)

Solution:

(a)

f(t) =



t 0 ≤t < 1

et 1 ≤t < 2

0 t ≥2

The graph is sketched in figure 1.

Figure 1. graph of f(t)

To find the Laplace transform of f(t), rewrite f(t) as

f(t) = t + (et −t)u1(t) −etu2(t)

L{f}= L{t}+ L{etu1(t)}−L{tu1(t)}−L{etu2(t)}

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