(x²+y2 ) prove ne du + y ax -2 sin2u ( Euler's Rule) 5. If u = cot-1 that x ду x+y

Q: State Leibnitz's theorem. If V1- x² y =sinx, then prove that (1-x) yn+2– (2n+1) xyn+1=n°yn= 0.

A: The Leibnitz theorem is used to generalize the product rule of differentiation. It states that if…

Q: 13. To evaluate f 3cos*x-1 dx. the integrand can be transformed to 1-sin2x a. [(3cos x+ csc²x)dx b.…

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Q: 1 29. J1+sin z cot z rp- sin z 30. -dr sec z 31.

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Q: If e /2i =sin2 then z = /21 - 2 In(sin2) + 4kTt – 2i In(sin2) – 4kt | 2i In(sin2) + 4k T - 2i…

A: We use the identity e^(2πki)=1 and by tanking ln both sides and then calculate.

Q: 1. sin (5x + ) sin ( x + T /2) %3D 2. tan (3x – T/2 ) = cot 2x 3. COs (x + п /2) sin ( T /2 – 2x )…

A: Given that three trigonometry equation  To find fundamental solution of the question

Q: 7. y = tan³ (2³) (A) 3x² sec(2³) tan(x³) (B)-3x2 sec(³) tan(x³) 8. y = cos(5x) (A) sin(5x) (B)-5…

A: Topic = Derivative As per guidelines, At a time only three sub-parts of a question can be answered.

Q: find the volume of the solid

A: Find volume of solid.Sketch the given function.

Q: sinh (iz) = i sinz sintiz) =isinh z,escia)= cosh: 2. !!

A: In this question, the concept of a trigonometric function is applied, Trigonometric Function The…

Q: @Hiw/ prove that cosiz=coshz @ Cosh'z -sinh'z=1 @1-Tanhz Sechz D cos hiz-l = cschz COS 3 =- Sinh z…

A: (2) we have to prove that,                                                      cosiz=coshz we know…

Q: Prove Sinhu.sinhv=1/2 (cosh (u+v)-cosh (u-v))

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Q: can you prove it, please ? cosh 2x=sinh^2x+1 = 2 cosh^2-1

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Q: iii) E(-1)"+1 cos²(÷) COS n=1

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Q: 1 dy sin¬' x If y = xv1 – x² + sin=1 x, prove that 24 – 2

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Q: b) sin 3x 3 sin x – 4sin3 x [Hint: sin 3x sin(2x + x)] Prove

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Q: Prove: sinh(x – y) = sinh(x) cosh(y) – cosh(x) sinh(y)

A: The basic relations we need to know to solve this problem are, sinh(x) = ex- e-x2cosh(x) = ex+ e-x2

Q: au at a²u 28. Solve Əx² U(x, 0) = 3 sin 2+x, U(0, t) = 0, U(1, t) = 0. = where 0 0 together with…

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Q: (1) If y = sin(tan-x), prove that %3D (1+ x²)?y" + 2x(1 + x²)y' +y = 0.

A: Since you have asked multiple questions in single request.so we will answering only first question.

Q: eiz-e-iz 1. Show that Sin z 2i

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Q: The integration expression sin(3In(x) + 4)cos(3In(x)+ 4)dx must be transformed into using suitable…

A: topic- indefinite integration

Q: 5. Verify whether the function f(z) Cauchy-Riemann equations or not. sinh 4x cos 4y + i cosh 4x sin…

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Q: Prove the identity sinh 2x = 2 sinh x cosh x

A: Prove that: sinh 2x = 2 sinh x cosh x

Q: The integration expression sin(3In(x) + 4)cos(3ln(x) + 4)dx must be transformed into using suitable…

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Q: 3. = Cos xsin +xax+a 5/2 3/2 1. Ans: Y= 4sin'x* sinx+ To CAx+4)^- Cax+a) 3.

A: To solve: dydx=cos3x sin2x+x2x+1 Solution Given differential equation can be written as: dy=cos3x…

Q: Š 4 sin(3t) 2+ cos(3t) 7 sin(3t) + dt (2+ cos(3t))²

A: Definite integration

Q: Classify and Solve; +9 = cos² t + sin² t + cosh t + sinh t + 3 sin t – 3t dt dt2

A: d2ydt2+9dydt=cos2t+sin2t+cosht+sinht+3sint-3t…

Q: 3. On (-∞, ∞), three functions y₁ (x) = cos 2x, y₂(x) = sin² x, y³(x) = cos² x are linearly…

A: Note: As per the guidelines we are supposed to only one answer because this is not a subparts…

Q: Find y' where y = 8 cos-1 (sin (19 z)) – cot-1 (14 z) Answer: 14 y' = -152 1+196 z2 -152 cos(19 z) +…

A: This question belongs to derivative of inverse trigonometric function.

Q: ♥ дzu ахбу = сезх созу, Ans: u(x,y) = sinx siny + x² + 2 cosy - #2² X 1 бу исx;#) =X бx U(x,п) =X,…

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Q: Verify the statement of Example 2. Also verify that y = cosh x and y = sinh x are solutions of y'' =…

A: Given- y=coshx and y= sinhx. To verify- y''=y Formula used- ddxcoshx=sinhx and ddxsinhx=coshx.

Q: Hico/ prove Thall Cosh2: - 2 tañ z Sinhz: 1+2 Tanhiz =

A: Since you have asked for a question with multiple sub-parts , we will solve the first three…

Q: What is the integrable form of / sin-c cos-sin@ do? 2 -2 | sinu du 2 | cosu du 1 – u² du u² – 1 du

A: In this question we have to find Integral form of the given Integral.

Q: 7. Prove: (a) sin iz = i sinh z. (b) | cos z|? = cos² x + sinh° y where, z = x + iy. COS

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Q: 3) Find the expention of sin³ 0 4) Proof that: cosh iz = cos z 5) Prove that: cosh² z - sinh²z = 1…

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Q: я/4 4 sin (4t) dt. 9- cos (4t) Evaluate the integral я/4 4 sin (4t) dt%3D 9- cos (4t)

A: Use integration by parts

Q: Quèstion 3 8+11x dx = 8-11x %3D 1 1 1 O A. 2 + C ++11x 8-11x 11x OB. tan-1 8 64 121x + C |- 1 11x 8…

A: Multiply by sqrt(8+11x) in denominator and numerator.

Q: Proove: Yn- n-1 x

A: Given That : Let us assume that  y=xnlog(x) Prove that : yn=n!x using Leibnitz  theorem

Q: Use Cramer's rule to solve for x' and y' in terms of x and y. x = x' cos 0 – y' sin 0 y = x' sin 0 +…

A: Solution is given as below:

Q: 7.2 (A) 1. S sin x dx cos x + C because(-cos x +C) = %3D dx 2. S cos x dx = because sin x 3. S tan x…

A: We need to solve the given integral. This can be done as follows:

Q: 30 csc (3x) cot(3x) dx (Use Substitution Method) 30 30 + C 6 cos² (3x) 6 30 30 + C 6 sin²(3x) 6 30…

A: Given that  I=∫ 30csc23xcot3xdx We need to substitute the appropriate value so we can solve it…

Q: Əz and Əx az #4. Calculate given that ду x² sin (z) + (x² – 1) y* = 2°.

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Q: What is the integrable form of sin-cos-sine de? 2 2 V1-u? du -2 sinu du u² – 1 du 2/ cosu du

A: We have given integral Let, I=∫sinθ2cosθ2sinθdθ As, sin2x=2sinxcosx sinx=2sinx2cosx2 So,…

Q: What is the integrable form of sin 0 sin-cos-sine d@? 2 2 u² – 1 du 1– u² du 2 | cosu du -2 | sinu…

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Q: If e/½i =sin2 then z = - 2i In(sin2) + 4k TI 2i In(sin2) + 4k T - 2i In(sin2) – 4kT -2 In(sin2) + 4k…

A: We will find the solution

Q: a3z cos(x – 4y) show that a3z 6. For z = дудхду

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Q: 11 csc²(3x) cot(3x) dx =, (Use Substitution Method) 11 11 +C 6 sin (3z) 6 11 11 +C 6 sin (3z) 6 11…

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Q: Prove the identities Then use them to show that a. sinh 2x = 2 sinh x cosh x. b. cosh 2x = cosh²x +…

A: Here we have to use the given identities to prove.

Q: 26 csc (2x) cot(2x) dx (Use Substitution Method) 26 26 + C 4 sin (2x) 4 2 26 26 + C 4 sin? (2æ) 4 26…

A: Integration.

Q: sin21/3x cos3 2/

A: