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EXERCISES - WEEK 5 1. Determine whether each integral is convergent or divergent. Evaluate those that are convergent. (a) 2 1 1 (2 1) dx x +∞ + (b) 0 1 3 4 dx x -∞ - (c) 2 6 1 x dx x +∞ -∞ + . 2. Use the Comparison Theorem to determine whether the integral is convergent or divergent. (a) 1 1 x e dx x +∞ - + (b) 2 1 t dt t e +∞ + (c) / 2 0 sin dx x x π 3. Find the values of p for which the integral converges and evaluate the integral for those values of p . (a) (ln ) p e dx x x +∞ (b) 1 0 ln p x x dx 4. The average speed of molecules in an ideal gas is 2 3/ 2 3 /(2 ) 0 4 2 Mv RT M v v e dv RT +∞ - = where M is the molecular weight of the gas, R is the gas constant, T is the gas temperature, and v is the molecular speed. Show that 8 . RT v M = THE UNIVERSITY OF DANANG - DANANG UNIVERSITY OF TECHNOLOGY APECE (Advanced Program in Electronic & Communication Engineering)
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Unformatted text preview: ---------------------------------------------------------- LECTURE ON CALCULUS WITH ANALYTIC GEOMETRY II 5. If f ( t ) is continuous for t ≥ 0 , the Laplace transform of f is the function F defined by ( ) ( ) st F s f t e dt +∞-= ∫ and the domain of F is the set consisting of all numbers s for which the integral converges. Find the Laplace transforms of the following function. ( ) t f t e = . 6. Find the value of the constant C for which the following integrals converge and evaluate the integrals for those values of C (a) 2 1 2 4 C dx x x +∞ - + + ∫ (b) 2 1 3 1 x C dx x x +∞ - + + ∫...
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