TABLE OF INTEGRALS: 1. 2. 3. [u 4. une du une 5. [u - n n...

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TABLE OF INTEGRALS: 1. 2. 3. ・[u² 4. une" du une" 5. [u² - n n [un-1 [eªu sin(bu) du = au un sin u du =-un cos u+n un-leu du + C [eau cos(bu) du = un cos u du = u" sinu - n · Sun-1 un-1 cos u du + C 2 / 1²-1 un-1 sin u du + C eªu[a sin(bu) — b cos(bu)] +C a² + b² eau [a cos(bu) + b sin(bu)] a² +6² + C Show clearly the steps you have taken and state the table entry number you use. Also, please show the open and closed intervals for part b when graphing. Thank you. 4. (a) Find f(t) = L-¹{¹(1+0=40) } . {. (b) Let f(t) = (2 – t)h(t – 2) – (3 − t)h(t − 3) + t h(t – 4). Graph f(t) for 0 ≤ t ≤ 5. Clearly explain how you obtained your graph. Common Laplace transforms (the ones we will use from Table 5.1) n! gn+1 2. L{t"} = 4. L{sin (wt)}: 6. C{sinh(wt)} 1. L{h(t)}: = 1 S n = 0, 1,... s² + w² 82 لا 3. L{eot}: (4) Antiderivative: {[(7) dr} = F(8) 8 = 1 s-a 5. L{cos(wt)} 7. L{cosh (wt)} (5) n-th derivative: L{f(n) (t)} = 8"F(s) — 8-¹ f (0) 2-1 - 14. L{h(ta)}: S s² +6² Five operational properties for Laplace transforms F(s) = L{f(t)} (1) Linearity: L{c₁f₁(t) + c₂f2(t)} = c₁L{f₁(t)} + C₂L{f2(t)} (2) First shift theorem: L{eat f(t)} = F(sa) (3) Second shift theorem: L{f(ta)h(ta)} = e-as F(s) е 2 ـ 82 8 e-as 8 f(n-1) (0) n = 1, 2,... • n = 1: L{f'(t)} = s¹F(s) — sºf(0) = sF(s) - f (0) • n = 2: L{f"(t)} = s²F(s) — s¹f (0) - sºf'(0) = s²F(s) - sf (0) - f'(0) TABLE OF INTEGRALS: 1. 2. 3. ・[u² 4. une" du une" 5. [u² - n n [un-1 [eªu sin(bu) du = au un sin u du =-un cos u+n un-leu du + C [eau cos(bu) du = un cos u du = u" sinu - n · Sun-1 un-1 cos u du + C 2 / 1²-1 un-1 sin u du + C eªu[a sin(bu) — b cos(bu)] +C a² + b² eau [a cos(bu) + b sin(bu)] a² +6² + C Show clearly the steps you have taken and state the table entry number you use. Also, please show the open and closed intervals for part b when graphing. Thank you. 4. (a) Find f(t) = L-¹{¹(1+0=40) } . {. (b) Let f(t) = (2 – t)h(t – 2) – (3 − t)h(t − 3) + t h(t – 4). Graph f(t) for 0 ≤ t ≤ 5. Clearly explain how you obtained your graph. Common Laplace transforms (the ones we will use from Table 5.1) n! gn+1 2. L{t"} = 4. L{sin (wt)}: 6. C{sinh(wt)} 1. L{h(t)}: = 1 S n = 0, 1,... s² + w² 82 لا 3. L{eot}: (4) Antiderivative: {[(7) dr} = F(8) 8 = 1 s-a 5. L{cos(wt)} 7. L{cosh (wt)} (5) n-th derivative: L{f(n) (t)} = 8"F(s) — 8-¹ f (0) 2-1 - 14. L{h(ta)}: S s² +6² Five operational properties for Laplace transforms F(s) = L{f(t)} (1) Linearity: L{c₁f₁(t) + c₂f2(t)} = c₁L{f₁(t)} + C₂L{f2(t)} (2) First shift theorem: L{eat f(t)} = F(sa) (3) Second shift theorem: L{f(ta)h(ta)} = e-as F(s) е 2 ـ 82 8 e-as 8 f(n-1) (0) n = 1, 2,... • n = 1: L{f'(t)} = s¹F(s) — sºf(0) = sF(s) - f (0) • n = 2: L{f"(t)} = s²F(s) — s¹f (0) - sºf'(0) = s²F(s) - sf (0) - f'(0)

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1 Sune do une 2 Susando S 4 Sune du un le SD SHUAN cos u du ... View the full answer