Show using Residue theorem: 1 2πί Is equal to pc+ix c-ix where n = rn sin no (21) 2a (1+2rn cos no + r²n) π |2|= sin po sin 2pa 2a dp kis or 2a = -. n

Q: (d) cos(z) sin(z)

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Q: Q4:B/Simplify [cos 20-i sin2018[cos 30+ i sin301² [cos 40+ i sin40]³[cos 20-i sin2015 using De…

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Q: c) fsin-¹(x) dx -1

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Q: Evaluate: /sin(/x)dx A 2/x sin(x)-2cos(/x)+C B2/xsin(/x)+2cos(/x)+C 2sin(/x)+2/xcos(/x)+ C D…

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Q: a 1 - x²) cos(nx) dx F(x 一九 4 sin(nл) — 4nл соs(nл) — 2n²л² sin(n) πη3

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Q: 3i +2V2 - 5n -4i +6(1 +a T/4 [(3sec2 t)i - (5 + sin t)j - (6sec t tan t)k)]dt Find 0. 2 2-57T-4 3i +…

A: ∫sex2xdx =tanx+c∫(constant)dx=x+c∫sint dt =-cost +c∫sect tant dt =sect+c tanπ4=1cosπ4=12secπ4=2

Q: 10. Using Cauchy's Residue Theorem, evaluate -2π (cos 30)² 1-2p cos 20+p² d0,0 < p < 1.

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Q: Ex:. Find the fourth roots of Z=-di Say no 4 K = 0₁12₂3

A: given z=-8i To find the fourth root of the given complex number proceed as follows comparing the…

Q: Use the Residue Theorem to show that sin²x ㅠ f² = 0 x2 dx = FIN 2

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Q: 5. Evaluate: 34 l vo dx dx. √9-16x2 0

A: To find the value of given integral.

Q: Q1: Determine -8 sec 7x (-4 cos? 7x + 6 tan 7x) dx. 32 sin 7x – 7 48 sec 7x + C 7 A -- В -224 sin 7x…

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Q: To show that rπ/2 sin" x cos" x dx: (m - 1) (m- 3) ... × (n – 1) (n - 3)... (m + n) (m + n - 2) (m +…

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Q: cos(3rk) In k 5k k 1

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Q: 4. Use residue theory to calculate the following integrals: 3x (a) P.V. · Loo (2²+49) (2² - 2x + 5)²…

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Q: Consider solving x²V1-x² dx using trigonometric substitution. Which of the following is true? a. 2…

A: In trigonometric substitution method, the variable is substituted with a trigonometric function to…

Q: The S- 3 dx is 5 sin 6x

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Q: a) _*? (1-x²)Z x+3 b) S- x2+2x+5 c) Find the value of sin(tan-1- sec-d) dx d) S x² Vx2+1 e) S Vx2+2x…

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Q: ( sin(3x+1) cos?(3x+1) Evaluate

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Q: /2 1. (a) Use the Wallis cosine formula to evaluate cos' x dx. (b) Evaluate | 1 - dx by using…

A: Wallis's formula state that 1. if  n is odd and n≥3,then ∫0π2cosnxdx=23.45.67 . . . n-1n Therefore,…

Q: Integrate 0 1 S √1 1-4x2 dx using a trigonometric substitution. sin-¹ (2x)√1-42²+ C 0 1 -sin-¹ (2x)…

A: The given integral is sqrt(1-4x2).

Q: Evaluate cos (x (x2 – 2x + 1) (2x - 2) sin (x 2 - 2x +1) (2 – 2x) sin (x? – 2x + 1) O (2x 2) cos (x-…

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Q: 5. y + 4y = sin x + 3 cos x Ans: y = C₁ + C₂ cos 2x + C3 sin 2x + sinx-cosx

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Q: The f sec$30 sin 30 d0 is {A} sec*30 + C 12 {C}-cos*30 + C {D}cos 30 + C 12 {B} sec 30 + C 18 18

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Q: 4. (i) Given that z = ete, show that z" +z- = 2 cos ne. Use this result to express cos$e in terms of…

A: 4i)

Q: Sz sin2m-10 cos2n-10 —— В (т, п) 2

A: It is known that, βm,n=∫01xm-11-xn-1dx

Q: Which of the following is a possible equation for the following trigonometric function?

A: To find which of the given equation is possible equation for the trigonometric function.

Q: Evaluate L. (4² COST (x² + a²)² da, where a > 0, using the residue theorem.

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Q: Find the root (3 – 3i)2 - + V18[cos(r/8) + isin(x/8)] O +V18[cos(x/8) – isin(t/8)] +3 V2[cos(1/4) –…

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Q: Q4:B/Simplify [cos 20-i sin20][cos 30+ i sin3012 [cos 40+ i sin40]³[cos 20-i sin2015 using De…

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Q: Can you please help with me (a)

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Q: Q4:B/Simplify [cos 20-i sin20][cos 30+ i sin301² [cos 40+ i sin40³[cos 20-i sin2015 using De…

A: Formula Used-  De Moivre's theorem: cosθ+isinθn=cos nθ+isin nθ cos mθ+isin mθcos nθ+isin nθ=cos…

Q: The square root of (1 2i) is (1) V5 (cos (+ 2n)+ i sin (5 + 2n")) for n = 0,1 COS (2) V5 (cos (+ 2)…

A: The general form of a complex number is given by z=a+ib, where a,b are two real numbers and i=-1.…