Prove (L1) (the linearity and additivity rule for Laplace transforms) using integrals. (L1) a, f (t) +

Prove (L1) (the linearity and additivity rule for Laplace transforms) using integrals.

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(L1) a, f (t) + a₂ f₂ (t) +...+ a f(t) (L2) f(ct) (L3) 1 (LA) t (L5) teat (L6) " (ne (0,1,2,...}) (17) h(t-to) f(t-to) = { (18) eªt f(t) (19) f'(t) (L10) f"(t) (L11) (412) f(") (t) (n= {1,2,3,...}) 0 f(t)dr (L13) et (L14) et- eb (L15) sin(@t) (L16) cos (@ t) f(t-to), t2to 0, t< to (L17) et sin (@ot) (L18) at cos( t) (L19) 8 (t) ("Dirac delta function") a₁ F (s) + a₂ F₂ (s) + ...+ a F (s) 1 S 1 s² 1 (s-a)² n! s+1 elos F(s) F(s-a) SF (s)-f(0) s² F(s)- sf (0) - f' (0) s" F(s) - F(s) S 1 s-a a-b (s-a) (s-b) (0) s² +0 S s² +0 (0) k=1 1 S-k f(-1) (0) (s+a)² + 0 ² s+ a (s+a)² + ² (L1) a, f (t) + a₂ f₂ (t) +...+ a f(t) (L2) f(ct) (L3) 1 (LA) t (L5) teat (L6) " (ne (0,1,2,...}) (17) h(t-to) f(t-to) = { (18) eªt f(t) (19) f'(t) (L10) f"(t) (L11) (412) f(") (t) (n= {1,2,3,...}) 0 f(t)dr (L13) et (L14) et- eb (L15) sin(@t) (L16) cos (@ t) f(t-to), t2to 0, t< to (L17) et sin (@ot) (L18) at cos( t) (L19) 8 (t) ("Dirac delta function") a₁ F (s) + a₂ F₂ (s) + ...+ a F (s) 1 S 1 s² 1 (s-a)² n! s+1 elos F(s) F(s-a) SF (s)-f(0) s² F(s)- sf (0) - f' (0) s" F(s) - F(s) S 1 s-a a-b (s-a) (s-b) (0) s² +0 S s² +0 (0) k=1 1 S-k f(-1) (0) (s+a)² + 0 ² s+ a (s+a)² + ²