

please prove part (b) use complex analysis and calculus of
residue
-dx neif a> 0 5. (a) x2+1 (b) For any real number a > 0, cos x dx ne"/a. a Hint: This is the real part of the integral obtained by replacing cos x by e
5. (a) Show that the following improper real integral is absolutely convergent cos 2x dr I (1+?}% " (b) If CR is the semicircle of radius R in the upper half plane with centre at z = 0, show carefully that e2iz lim R00JCR (1+ z2)2 d% = 0 (c) Use residue calculus to evaluate the real integral I of part (a)
5. (a) Show that the following improper real integral is absolutely convergent cos 2x dr I (1+?}% "...
1. Using the identity: eimø = cos(m) + i sin(m) show that m is real and equal to an integer.
4. Evaluate the following integrals using the Residue Theorem. Justify your calculations, show the work. (10 points each) a) 12 cos a + 13 2 da b) (x2 +6x + 10)2 x sin 2x 24 13 da: c)
4. Evaluate the following integrals using the Residue Theorem. Justify your calculations, show the work. (10 points each) a) 12 cos a + 13 2 da b) (x2 +6x + 10)2 x sin 2x 24 13 da: c)
15. Using that sin' (2) = cos(x), cos' (2) = - sin() show that arccot (0) = 1 +22
Solve 8 sin? (w) – 10 sin(w) + 3 = 0 for all solutions ( <w < 27 W = Preview
Using Integral Calculus
b * W- Sa f(x) dx using Integral calculus 4. A rope is being unwound and the force of gravity on the rope is 12.0 N/m. When 20 m have been unwound, how much work is done by gravity to unwind another 30 m? (6 marks)
Please answer with all steps. Thanks
Given "x4 3 cos + 7 sin t 0.75_dt F(x) =let, d, G(x) =1 dt 5t0.75 0 Using the Fundamental Theorem of Calculus Part II, calculate the limit Lim
Given "x4 3 cos + 7 sin t 0.75_dt F(x) =let, d, G(x) =1 dt 5t0.75 0 Using the Fundamental Theorem of Calculus Part II, calculate the limit Lim
Given that W = - Spdv and pvy = constant for an adiabatic process, show using differential and integral calculus that W=-S, pdv gives on integration
1. Let ab and f E C[a, b], and let E(0, ))- - (co +c)w(a) da for some weight function w(x) >0. (a) Use calculus to write down a linear system for the critical point of E(co, c1). (b) Is the solution of this linear system the same as that of the normal equations arising from the use of Theorem 2 on page 395 to optimize co, ci under the norm 1/2 ? (c) Use your results to find the...