Show that sin mx cos nx dx = 0 for positive integers m and n: i using the primitive, ii using symmetry arguments.

Q: 20 a Show that sin (A + B) + sin (A – B) = 2 sin A cos B. b Hence find: i 2 sin 3x cos 2x dx ii sin…

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Q: Use symmetry to evaluate the followingintegrals.

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Q: Q6:-Prove that: y" =1 y =csc(tan"") if

A: Given y=x4·csctan-114x2-1 Let tan-114x2-1=α ⇒tan(α)=14x2-1 ⇒cscα=2x ⇒y=x4cscα⇒y=x4(2x)⇒y=x22

Q: Use the substitution t = tan(x/2) to find the integral: 2 1+2 sin a dr Cos I

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Q: If x = t + 2 sin(at), y = 3t – cos(nt), evaluate at t =;

A: Given that x=t+2sinπt, y=3t-cosπt, we have to find dydx at t=12. Now, dydx=dydtdxdt…

Q: Show that 0, if m if m + n sin(nx) sin(mx) dx T, = n IT

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Q: If x=sin@,y=0 ,y=cosp0,Prove that (1-x²)y,-xy,+p°y=0

A: Given, x=sinθy=cospθ Required to prove, 1-x2y2-xy1+p2y=0...1 Then finding dydx by using chain rule,…

Q: Use symmetry to evaluate the followingintegrals.

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Q: Suppose z and y are funetions of time and satisfy the formula e sin y = In y + tan dy Write a…

A: Solution:- differentiate with respect to t.

Q: = cos(x + y), when x = 0, y = " %3D

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Q: if y=tan x,prove that y'''=2(1+y²)(3+y²)

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Q: Let z = 8(cos 50° + i sin 50°) and w=2(cos 30° + i sin 30°). Express in the form r(cos+ i sin 0):…

A: Here we can use the following formula to find the results eiθ=cos θ+i sinθande-iθ=cos θ-i sinθ

Q: The integral (cos(3x)) sin(3x)dx under the change of variable u = cos3r becomes u du. Select one: a.…

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Q: •2n Let a and b be distinct nonnegative integer. Prove that Sa sinax cos bx dx = 0 %3D

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Q: 9. Let r = 3 sin tî + 3 cos tj + 4tk. Calculate N

A: Given r(t) To find N(t)------ unit tangent vector

Q: Find the residues of sin(2²) at z = 0 and z = π. (sin z)2

A: To find the residue of sinz2sinz2 at z=0 and z=π Let  fz=sinz2sinz2=sinz2csc2z…

Q: Q.3- Prove that: 2n 1 (cos =0 +i sin-) + 10 2cosne eino

A: The given problem is to prove the following complex identity. (cos n/5 θ +i sin 2n/10 θ)+1/einθ =…

Q: Prove the left side equal to right side in the following: s2 (eice = (sin at + at cos ar] 1 L-1 (s²…

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Q: D[Expf-3x] Cos(nt], {x,2},t]

A: Correct option is D

Q: Use the Substitution Rule to provethat∫cos2ax dx =(x/2) +(sin(2ax)/4a) + C.

A: The given integral is ∫cos2axdx=x2+sin2ax4a+C

Q: Use symmetry to evaluate the followingintegrals.

A: We have to find definite  integral using symmetry.

Q: Prove: L{cos at} = s2+ a?

A: We will prove the given statement.

Q: Ex2: Use the identities 4 and 5 to prove that: cos(x- y)-cos(x+y) (a) sin x sin y = 2

A: Formula used -  cos(A-B) = cosA cosB + sinA sinB  cos (A+B) = cosA cosB - sinA sinB

Q: By the samc way, prove that g 7 sinw.k MUIS Z N-27 cesW. +1

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Q: 18.) Suppose x = R cos(0) sin($), y = R sin(0) sin($) and z = x² + y? + z? = R²Y R cos(). Verify the…

A: Given

Q: If y = tan x, prove that y``= 2(1+y² )(3+ y²).

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Q: D. Use the Wronskian to detcrminc whether or not the vcctors arc independent or not (the answer for…

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Q: DIfferentiate the following fuh ction with respect to K In X %3D Sin a

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Q: 7/2 79. Prove that 0< sin x dx < Ja/4

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Q: Q9: If F(x, y, 2) = [sinh-(sin- (In (/sin(cos(x²)) y? + zyx + y, find and ду az Q10: If z =…

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Q: 2-If w(x, t) = Sin(x - + 3t) + Cos(2x+ 6t), prove that wtt = 9wxx %3D %3D

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Q: The integral S (A S- (в x dx x² + √√√x²-1- sino do 1 + cos sec²0 do 1+tano " cose de 1-sino sece…

A: Use trigonometric substitution

Q: Determine c, and c2 so that y(x) = c, sin 2x + c2 cos 2x + 1 will satisfy the conditions y () = 0…

A: Consider the function: yx=c1sin 2x+c2cos 2x+1. Calculate c1 and c2. Given yπ8=0 and y'π8=2.   Since…

Q: If y= tan x , prove that y`= 2(1+ y² )(3+ y²).

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Q: ,In sec2(sin2y)dy can be evaluated by writing it as sec2y (A @ In sec2( sin2y) · cos2ydy ®/…

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Q: /Prove that the value of cos()dz, when z = T + 2i is e + e-1

A: Complex integration of the function

Q: ) Given that g(0)- 2sino - Scost, express g(0) in the form R sin(0 -A). giving values of R.

A: As per bartleby guidelines for more than one question asked only one is to be answered please upload…

Q: Show that r = cos(2pi) is symmetric about the y-axis.

A: The solution of the problem is given in the next step.

Q: (a) If u(x, y) = cos-1*" + y* ) x² + y? = COS then prove that ди du + y = -2cotu

A: Given: ux,y=cos-1x4+y4x2+y2

Q: 0, o° s 0 < 360°, given the follo Cos e = and 0 in QIII 2.

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Q: Show that J, (z cos0)d0 = - R/2 os e)de = 1-cos z %3D 21

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Q: Q/ IF y = Sim ܠܐ 10²² (4) y = 3/² sin(x) cos(x) Prove that

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Q: Determine c1 and c2 so that y(x) = c1 sin 2x + c2 cos 2x + 1 will satisfy the conditions y(pi/8) = 0…

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Q: Q6:-Prove that: y" = 1 y =csc(tan"" ) if

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Q: Find sin zj and cos ( – nj) in the form u + jv.

A: Brief answer is given below.

Q: Sketch X,(t) and Xa(t) Where X,() = - Sim3t Sin3t %3D 3. X,(e) = /a Sin 3t %3D

A: To draw the graph of functions X1(t)=-13sin3t and X2(t)=13sin3t, substitute different values of t in…

Q: dz az Use the chain rule to find and as sin(8) r = st, 6 = s + t3 = az %3D as az %3D

A: Given:- z=sinθrr=stθ=s3+t3 To find:- ∂z∂s and ∂z∂t