If I(x) = | u"-1e-" du, therefore 0, r°(x) = (u*)*-e" du du -u? 0, O Option 1 r°(=) = (u")** e " du du -2u 0,

Q: 3. Verify that ysin x-xy =0 is a solution of the equation y(cosx-y)dr+(sin x- 2xy)dy =0.

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Q: 81 fx, y) dy dx 10 10 O-10 10 1아 1아 -10 10 -10 -10 81 -x F(x, y) dy dx = f(x, y) dx dy

A: we have to change the limit of integration.

Q: 2 tan(4x-1)-3 cos² (4x-1) If we let u = 4x – 1 for the dx, what will be the resulting integrand?…

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Q: 2 sin? 11. y = 3x 2x2 12. y = cos°V2x – 1 3. 13. у %3D Vtan(х?)

A: As we know that; The chain rule: fgx'=f'gx·g'x The Quotient rule: fg'=f'·g-g'·fg2 The Power rule:…

Q: 1 If z = 2x2 – 3y with u = x² sin y and v = 2y cos x, determine expressions %3D az dz for and du

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Q: Differentiate y = In (sin + cos ) cos -sin 2r Va(sin a + cos V") sin E + cos E 2r V(sin a + cos a)…

A: we will apply, chain rule: ddxf(g(x))=ddg(x)(f(g(x)))×ddx(g(x)) Sum rule:…

Q: 2. A particle moves along the parabola y = x in the first quadrant in such a way that its…

A: Given: A particle moves along the parabola y = x2 in the first quadrant in such a way that its x -…

Q: 4. Find fyryzyaz, if f(r, y) = x² cos[100(e" + /1+ y²)®]

A: Given; f(x,y) = x2cos[100(ey+√(1+y2))y]

Q: tan x dy +y sin x dr = 0, when x = 0, y = | O y (3- cos 21) = 2 O y (cos 2x O y° (2 – cos 2x) = 1…

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Q: 1- d/dz [cosz] = -sin z

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Q: r(10,3,3) dx (1,1,1) dy dz. - -

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Q: 」enr一2) dz csc (4r – z) dz dz dz 17. 2 V1 – sen 2z V1 – cos (4- – 2z) In | csc (4- – z) – cot (†7 –…

A: Integrate as general

Q: y = 46 tan B3 – 2 = 4¢ tan az – 2 %3D and y = 46 tan B2 +2 = 46 tan æ3 + 2 %3D and hence all the gas…

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Q: 7,9,-4) y dx +x dy +3 dz = (5,4,1)

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Q: S cos(3z)sin1º(3z) dz C 1 (cos (3z))10 + C 3. (sin(3z))10 + C D 1 (cos (3z))1 + C 33 (sin(3z))'l + C…

A: Topic = Integration

Q: 4  find The indefinite integral

A: To evaluate the given integral

Q: sin(6y) sin(8y) dy.

A: Topic - integration

Q: Step by step

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Q: 2. +4y= sin(3t), y(0)=2, y'(0) = 0

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Q: 3x cos 8x dx e3x e3x A) [8 sin 8x - 3 cos 8x] + C B) [8 sin 8x + 3 cos 8x] + C C) (8 e3x sin 8x + 3…

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

Q: Differentiate: y = In cosh (cos 2) O y' = 6 sin 2z tanh(cos 2z) y' = -6 sin 2a coth(cos 2x) Oy = -6…

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Q: (2) If y = (sin-1(x))², prove that (1- x2)y" – xy' - 2 = 0.

A: here first we will derivative with respect to x then squaring both side and put value of the y then…

Q: 3) Show how the substitution u = 2 x shows that - - t/2 -Tt/2 S sin"x dx = S? cos"x dx

A: Let u=π2-x Differentiating u w.r.t. x- dudx=-1du=-dx Considering left hand side of the given…

Q: Find dy if y= 5x² sin4x dr O None -20x? cos4x – 10xsin4x 20x² cos4x + 10xsin4x O 10x² cos4x +…

A: We apply u*v rule of derivatives d/dx (u*v) = v*du/dx + u *dv/dx

Q: then what is y at x=1?

A: We can find y at x=1 in the following steps.

Q: tan '(z) Find for y = In( - dy dz You do not have to completely simplify!

A: topic - derivatives

Q: Intergrate:

A: To evaluate the integral.

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: (2) Show that y=x sinx is a solution to y"+y= 2 cos x.

A: To Show: y=x sinx is a solution to y''+y=2cosx

Q: Solve the problem: u – kuxx = e-t sin x , 0 0; satisfying the conditions u(0,t) = 2n, u(t, t) = 4n,…

A: Let the transformation ux,t=vx,t+Ax+B ...............(1)uxx=vxxut=vtthe problem ut-Kuxx=e-t sinx…

Q: 1. Use Fermat's principle to find the path followed by a light ray if the index of refraction is…

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Q: S sin x coszxdx = cosix + c cosix + O Option 2 O Option 3 -cosix + e -cosix + c O Option 4 O Option…

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Q: Show that y1(t) = cos(3t) and y2(t) = sin(3t) are linearly independent for all t

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Q: 2. A. If : = sin(3x+2y) , verify that 2 -3 = -6z ax2 B. If f = x³y3 + 2xcosy + y?lnx then show that…

A: x=sin (3x+2y)

Q: z sin(2- 32)+cos(2 - 3a)+c | 4x cos(2 – 3x) dx 4x cos(3 3r) dx Option 1 Option 2 |4 cos(3- 3z) dr…

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Q: 4. [(3sin 2f – 3 cos 5t )dt 5. sin x dx cos x

A: 4) Given:            ∫3sin2t-3cos5tdt       =∫3sin2t dt-∫3cos5t dt taking constant out…

Q: 8. 4 6° sin°3ycos *3y dy

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Q: 35. Given that d = 6(sin 2x - cos 3x) and y = 0 dx when x = 0, y = A -3 cos 2x - 2 sin 3x – 3 B 3…

A: We can solve this problem using initial value problem.

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: Suppose that the position of one particle at time t is given by X1 = 5 sin(t), Y1 = 2 cos(t), osts…

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Q: dy+チ=-

A: Note: Since you  have mentioned multiple question, I answered first problem. If you need answer of…

Q: 1. dz (1² + 5)² 2. COs z dz 3. sin 3r cos 5z de

A: Given: I1=∫x3x2+52dx, I2=∫excosxdx, I3=∫sin3xcos5xdx

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: A particular solution for y +2y +y=1-cosX

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Q: (4) find dy if 3って - ソ= e* (2Co83x +3Sin3x) e = Sin (x+3y).

A: Given 1. y=e2x2cos3x+3sin3x 2. e2x=sinx+3y

Q: 1. y = Arctan 3-V9-x2 2. y = 4 Arccos + 2v4 - x2 2 y = In (*+2), (x-1)2 3. у%3DIn

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Q: Suppose x and y are functions of a third variable t If sin2(x) + cos²(y) =- and dr dy what is the…

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Q: Differentiate y = rarctan(e") re arctan(e")+ re Option 2 Option 3 re arctan(e") - Option 4 None of…

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Q: 6. Vsin x sin 2xdx Vtan 4x 7. J cos² 4x 8. cos 2xsin 2.xdx |arctan(3 – x) - dx 9. 1+ (3-х)? 3e 2y…

A: Given

Q: 19. Sobve (1-x*)-+m*y=0 dy=m dx where x=0, y=0,

A: