1. Use Cauchy residue theorem to evaluate the following integrals. (a) √2do 5+3 cos 0 dx (b) (x²+1) 3 xdx (c) So 25+1 COS (d) f (x+1)dx
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- Use Cauchy's Residue Theorem to compute the improper integral (x + 3) sin(x) dx (x² + 1)(x² + 4)² and express your final answer in exact form. 8(a) 4. Evaluate each definite integral using the Fundamental Theorem of Calculus. ja (c) / (cos r – V3sin r) da (a) / (6 – x) da -4 3x² – xª + dr */6 (b) | (1+ sec² 0) do -1/6Solve the following showing clear steps
- Find the quadratic approximation of f at x = 0. ,sin 8x f(x) = e O A. Q(x) = 8x- 32x? O B. Q(x) = 1+ 8x + 32x2 OC. Q(x) = 8x + 32x? O D. Q(x) = 1 – 8x + 32x?3. Consider the definite integral x³e-2xdx. Fill in the blanks for the equivalent definite integral after making the variable substitution t = – 2x². Upper bound (1) New Function (2) dt Lower bound (1)Use the Residue Theorem to show that sin² x x² 1. 0 dx = ㅠ 2
- /2 1. (a) Use the Wallis cosine formula to evaluate cos° xdx. 3 (b) Evaluate | 1 -dx by using properties of improper integrals. 1/3 (х-2)"3a) Rewrite A = √=x² + 4x + 21 in the form A V 6² - (x − a)². b) Use a trigonometric substitution to evaluate the indefinite integral. S/ dx You must show all steps.1. Evaluate each indefinite integral and check the result with differentation. (a) / – - x-3/2)dr (d) / (sec² 0 – sin 0)do (x+ 3 dr (e) (tan y + 1)dy (c) / x + 6 (3 cos(r) + 2VT)dr