(1) a) Using the Cauchy Integral formula (or otherwise) compute the following integrals. i) S 121=4 ez Cosπz dz ii) S z²+2Z Cashz 12-21-22-1 dz dz S Lii) √ (2²+9) (z+9) 121=4

Evaluate the integral by completing the square and using trigonometric substitution. / Vx? – 4x – 45 dx - (Express numbers in exact form. Use symbolic notation and fractions where needed. Use C for the arbitrary constant. Absorb into C as much as possible.) 2(V+5)(z-9) (x+2) – 49 In( |×-2 1)) +c - 4x – 45 dx = (x- 2+ / (x+5) (x+9) Incorrect

Evaluate each of the following Indefinite and Definite Integrals: a) b) √6x³-18x² +7 dx 8 13-14 d) [√x-7√√x+17√√x dx. c) e) 12x³-9x² +2dx h) dx. f) (₁²-1)(4+31) dt. f²¹²-62² +42² -2 dz. 2y³-6y²

1) Evaluate the integral J 2) Evaluate the integral sec² t dt (tant - 1)² 8 dx 2 5 x√x² - 16 Hint: Use a suitable u-substitution. You may

Q) prove that -2P xSinx 2. P+1 Xcoshx (P²-1)? 3 P-1 -XcosX (P+1)? 4 2P X sinhx (P²1)?

6. Complete the square to get an equivalent expression: 7-6x – x² | No integration necessary on this one!

e²x b) Prove that, 5000 (@3x+1)2 dx = 2π 9√3

Evaluate the integral 4 (2a+4) (x2 + 4x + 2) da by making the substitution u = x2 + 4x + 2. x2 + 4x + 2 + C NOTE: Your answer should be in terms of x and not u.

Show that 2x? +8x+14 =2 (x+2) +3 by completing the square. Hence, evaluate the V3 integral dx. (x+2)/2x? +8x +14

The solution of some second order linear DEQ is exp(-7x)[5sin(1x)+ 7cos(1x)] which can also be expressed as A*exp(-D*x)*sin(F*x+G). Determine A,D,F, and G (degrees).