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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- How do I do #1?4. Evaluate the integral below by following a series of steps: 2x +3 J 4x² + 5x + 6 (a) Evaluate f dx by using trigonometric substitution. 4x2+5x+6 Hint: Pull a 4 out of the integral, then use "completing the square" in order to write the denominator as (x + A)2 - B 2. Then use a trigonometric substitution. 8x+5 (b) Evaluate f dx (Don't think too hard! The answer is easy!) 4x2+5x+6 (c) Find constants C and D which allow you to write the original in the following way: 8x +5 dx + D| 2x + 3 dx 4x2 + 5x + 6 dx = C J 4x2 +5x + 6 4x2 + 5x + 6 (d) Use parts (a), (b), and (c) to evaluate the integral!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)