Prove the following equalities using the residue theorem. Justify your steps. (a) √2 2π de 2πT 0 a + sin ¤ Fa² - 1^ Fa² = π a > 1; 0 (b) 8 x-a π So z +1dx = sin(sa) ୮ 1

d -cosh(2x² + 1). Evaluate dx A 4xsinh(2x² + 1) B 2xsinh(2x² + 1) C) xsinh(2x2 +1) D (2x2 + 1)sinh(2x + 1)

Suppose f (x) = sin(2– 3x). Evaluate: A - 3cos( 3x – 2) +C © -om(3t - 2) + C B © jcon(3r - 2) + C D Зсов ( 3х — 2) + С

#9a using Descartes's rule of sign

7x 18 - 1 r2 +1 sin x2 – 3x + 2x + Find the natural domain for f(x) =

| (15t 2 – 5t) cos (6t - 5t) cos (6t-1+ t²)

Write f(t) = 7e.8t in the form f(t) = Pob' and determine Po and b. (Give exact answers using symbolic notation and fractions if needed.) Po = b =

1 dx. x2-2x+8 Find I = Choose the factorization of the denominator from the following: Question 1) a. (z-1-iV7)(z+1+ iv7) b. (z-1-iv7)(z-1+ iv7) c. (z+1-iv7)(z-1+iV7) d. (z +1-iV7)(z +1+ iv7) e. (z - 1-iv8.5)(z + 1 + iVB.5) f. (z-1-iV8.5)(z - 1+ iV8.5) g. (z +1- iv8.5) (z -1+ iV8.5) h. (z+1-iv8.5)(z+14iV8.5) Question 2) Choose / from the following: a. 1.1417 b. 1.1874 c. 1.2349 d. 1.2843 e. 1.3357 f. 1.3891 g. 1.4447 h. 1.5024

¸¸ S'√ 16–16 x² d x = · (a) 16 π (b) π square unit. ©2π (d) 4π

kindly solve