A: limy-o sinx(1-cox) 2x

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Q: SEr up AN INTEGNAL KEPRESENTAT7 ON OF THE ARGA OFTHE REFGrON BOUNDEO BY y =x3,y=0, x=-l, X= 2 ド

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Q: Find the Leplace bransform. of Hhe fure tion ESinut octca to biven that

A: Laplace transform of periodic function: The Laplace transform of a periodic function f(t) of period…

Q: Discuss the Singulazities and Poles of fiz) = zéSinz (z-2) (zこいこょうと+ら)(2t (こか) +4

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Q: 2 1-X ie 2=tny then prove that da

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Q: dy is If y = In then cosh z

A: Given,          y=lnxcoshx

Q: Find fy if f(x, y) = sin 5x cos 5y O fy = 5sin 5x sin 5y fx = cos 5x cos 5y O fx = 5cos 5y cos 5x fx…

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Q: Consider the following IVP: у" +у'%3D-сos x;B у(0) %3D а, у(0) -b Find the general solution. A y (x)…

A: Solution of the differential equation

Q: Prove That:- o sinhz= loy (Z+lz*+l )

A: To prove, sinh-1z=logz+z2+1     Lets consider, y=sinh-1z…

Q: az əz 1. ax' dy If z = In(e* + e"), show that %3D %3D

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Calculate Hhe fourier coeffrecents Of the fancdion fR) over the nderval -スと×L2 here: Fx)=(0,-2くメCー…

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Q: Define stpes to obtain the general solution of a 2nd order homogeneous ODE with real coincident…

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Q: Page 4of 8 3. Evaluate the following integral or show that it is divergent: roo In x xp a) S" x+1 b)…

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Q: 2. Use the Definition of Residue to show that: dz -ni Izl=1 z² sinh z 3 Where 0 < |z| < T.

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Q: Q1. Verify Euler's theorem for the homogeneous funetion, fxy.2)-axy +byz + czu

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Q: Let C be the shortest path that connects z to mi and /= . (sinhz + 1)dz. Then |=-1-cosh t+ n(1-1)…

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Q: lim (z,y)→(0,0) (3 + (x² + y²) In(x² + y?)) 00 does not exist O 30 1,

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Q: tan-i z = ÷In 1-ja sinh-1z = In(z + vz2 + 1) cosh-1 z = In(z + vz² – 1)

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Q: and let ah -(-1'I Let In tan (a) for n = 0,1, 2 Intz 1) Show that In + Io+ り+! (-1)" for nal an-…

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Q: 9. Prove (1, sin z, cos z} is linearly independent in [0, 2x].

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Q: Evaluate the improper intergal 1 dx using the residue (x² + 16)² theorem.

A: We need to evaluate integral.

Q: (b) Compute r if: arctan r + arctan 1 = 2 ( arctan z arctan

A: tan-1x+tan-11=2tan-1x-tan-113

Q: 1. Show that sinz is unbounded on C and lyl <|sinz] <ebl z=x+yi € C. for

A: Given:         We have to show that sinz is unbounded in ℂ and y≤sinz≤ey   for z=x+yi ∈ℂ Function…

Q: 5 dx 8x (x+1)? 1

A: First evaluate indefinate integral

Q: et X~U([-, 1) and let y = tan(r), show that y Cauchy(0, 1)

A: # using Jacobian transformation we have proved that y=tanx follow Cauchy distribution (0, 1)

Q: y = sech*x = In(_+ V1- x², sechx = In(- 1+ V1-x² Prove that .... ..

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Q: 2.Prove that I (sinwt) : s2+w?

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Q: Use Greens Theorem to evalute fyx² dx + x²dy as shown below Leh Half of (0,5) +y-25 (0,-5)

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Q: Evaluate the integral S 3012 2/3 de using the two-point Gaussian quadrature rule. 1+16z4

A: Given integral is ∫−121221+16x4dx. We have to evaluate the given integral by using the two-point…

Q: 2. Use the Definition of Residue to show that: dz -ni 3 Izl=1z sinh z Where u> |z| > 0

A: Prove,∫z=1dzz2 sinh z=-πi3 ∫z=1dzz2 sinh z=∫z=1fz dz .........equation 1 fz=1z2 sinh z , 0&lt;z&lt;π…

Q: Evaluate the integral f 2x ¡dx using the two-point Gaussian quadrature rule. 16 \\frac{2}{3}\)

A: Given integral is ∫−222x216+x4dx. We have to evaluate the given integral using the two-point…

Q: 3- Let µr1 = 2 in region 1, defined by 2x+3y-4z > 1, while µ2 = 5 in region 2 where %3D 2x+3y-4z <…

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Q: Express the limit lim E (4 cos (272;) + 6)A¤; [2, 4] over as an integral. n- 00 i=1

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Q: Find the Selution of (1-t +3t)4 (t) di - , (2) In which IL) denoter Bessel function

A: Leibniz integral rule : ddx∫bxaxFtdt=∫bxax∂∂xFtdt+Faxddxax-Fbxddxbx   Linear Volterra Equation of…

Q: Evaluate the improper intergal dx using the residue 2 (x² + 4)² theorem.

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Q: Prove that arctan x + arctan y = arctan(x + y) / (1 − xy),  xy ≠ 1.                   Use the…

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Q: show that the. following Complex functron is dis continuous at z=i. fCZ) = スー! %3D 3.

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Q: lestion 25 Consider the following IVP: y" + 3y' + 2y = 12x?; y(0) = a, y'(0) = b

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Q: finde general Sonotion 1- y' = See'y 2X-1 %3D 2- 3-Xy=Y+2x $ in?

A: This question can be solved by applying the concept of differential equation.

Q: Find leng th of the Cardioid Vsa6 from e -oto0 - 1 %3D

A: Use the formula of the length of the polar curve. L=∫0πr2+drdθ2dθ Here, r=aθ3 and drdθ=ddθ(aθ3)=3aθ2…

Q: Verify that cos kx a solulion of Also, stafe what kind of DE is the DE above

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Q: D Find a singular saluhion of DE: %3D

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Q: let us efine fn enj-S (n) = otherwise b) we wo ld like to calco late the lebesgue integral Sf dm…

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Q: Show a complete soluton 2 3 CO-10y 10sin x

A: Given:

Q: Use residues to evaluate the integral 2T 1 de. %3D 5+3 cos 0

A: To evaluate : ∫02π15+3 cos θdθConsider unit circle |z|=1i.e. z=eiθ⇒dθ=dzizAlso, we use cos θ=z2+12z,…

Q: (B) Compute div F and curl F for F= x²yi- (z3-3x)j + 4y2k %3D

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Q: if :z = cos 0+isin 0 2z² z²+1 prove that: 1+itan 0

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