Question 2 Differentiate the following functions (i) If y= e-t sin 4t, prove that +6+2y = 0. (ii) Given that sin(x? +y) = y (3x + 1), show that dy 2x cos(x? + y) – 3y² !3! dx 2y(3x + 1) – cos(x² + y)

Q: 4. Show that sin(0 – ø) R2 – 2r R cos(0 – 4) + r² dø = 0. -

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Q: Use a change of variables or Table toevaluate the following indefinite integrals. Check your work…

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Q: Question 7 Find the partial derivative of z with respect to x 8: da = -2x (3x²+1) sin(x²) + 6xcos…

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Q: x sin wx dx x2 + 6x + 10 Show that (2 sin 3w – 3 cos 3w) w > 0 %3D - 2e2w

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Q: (cos 2y – 3x²y²)dx + (cos 2y – 2x sin 2y – 2x³y)dy = 0 |

A: We will find out the required solution.

Q: To evaluate 3cos x=1 dr. the integrand can be transformed to 1-sin2x A.) S(3cos³x + csc?x)dx B.) 3…

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Q: 1. Determine the general solution of this equation in terms of C ( sin y- ysin x) dx + (cos x + x…

A: We have the following solution for given ODE:

Q: Question 2 Find the differential dy: y= sin O dy E In a + sin( rVE dr Cos ( xY O (E) None of these…

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Q: y = (sin x +cos x) sec x = sec² x

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Q: If k 0 then the trigonometric integral [ 3 sin³ (kx) dx is equal to ¹(3 cos(kr) cos³ (kr)] + C

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Q: d sin x 1 dt| = dx et + 5 cO X 1 1 1 (A) esinx + 5 (B) e* + 5 (C) (esinx + 5) cos x (D) e sin x

A: We know that integration power rule is given as - Integration of xn is nxn-1.   Here i attached the…

Q: 1. f/² sin () cos (+) sin (²) d dx (Consider initially the first 2 terms)

A: We have to use the formulas for 2sin(A)cos(B)-sin(A+B)+sin(A-B) and 2sin(A)sin(B)=cos(A-B)-cos(A+B).…

Q: QUESTION 4 Suppose that x2 -v3 +2xy= 12. Use implicit differentiation as presented in Section 14.5…

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Q: Sf(x) dx = 1 0.694 (sin(x) dx = 1 0.694 COS TX T C = 2 C 0.694 How? ==→ 1-C05 (0.69471) = 1694 -

A: To determine the value of C .

Q: 11 Evaluate (( sin(y²)dydx where 2: 0<y<1, 0<x<y. 0 x a) 1-cos(x) b) 1-cos(x)) 1 - cos() 2 d) 1.…

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Q: x4 1 If u = tan(- + y° x² – y² - then show that ди du + Y- ду X - = sin2u dx

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Q: dz 1) Find ifz = 3 cos x – sin(xy),x =y = 3t - dt

A: NOTE*** As per bartleby guideline for more than one question asked only first is to be answered…

Q: (a) Show that cos(x²) > cos x for 0 ;. 1

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Q: 2. The suitable Trignometric substitution that solves fx(x-2) dx is (x-1)=sin 0 x-1)3sec0 (x-1)=tan…

A: We can find the answer as below.

Q: Question 4 Calculate the given integral: / tan (4x) dx 1 1 a) 능 -tan(4x) + 4x + C tan’(4x) + 12 1 b)…

A: We have to solve the integral

Q: 7(e)₁² X y= » - [1-171² 2 sec- y = (tan 17 1² ² X y = [cscx]arcsin (cos.x)

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Q: QUESTION 3 Find dy/dx if cos(xy) = 2xe2y + In(2y) by using implicit differentiation. %3D

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Q: Question 3 Calculate the integral: 2x sin(3 x) dx 2 2 a) Ox cos(3x) + sin(3x) + C 2 b) O-x cos(3x) +…

A: Apply integration by parts

Q: 4x /² sin () cos (+) sin (²) dx (Consider initially the first 2 terms) 3

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Q: Question 7 Calculate the integral: /x' cos (2.x²) dx 1 o- sin(2x³) + r cos(2r*) + C b) O sin(2r*)…

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Q: Question 3 * Q3) Simplify (if necessary) then Compute: y¨+ 2y Where y(x) – |,raz (z'x t cosx*)|…

A: Given: To calculate  y'' + 2y' Here,  y(x) = ∂2∂x∂zz2x + coz x4 4 (cosh 2x - sinh 2x)16cos 2π xddx…

Q: QUESTION 2 Determine a potential function for F(x,y,z) = sin(y)e³²i - xsin(y)e?j + 3xcos(y)e32k or…

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Q: 1. Solve the DE: (cos y- cos x)y' +cos y sin 2x = 0

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Q: Question 8 Integrate the function. dx V9 - x2 COs:1 X+C 2 cos x+ sin x+C D sin *-C Question 9 413

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Q: The solution of the DE (1+ tan x)y'=-secx coty is O None of these siny + In cosx| = c %3D sinx +…

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Q: 25. æ sec (a²) tan(æ²) dæ | sin? x dx (Hint : cos (2x) 26. 1 27. dx 4- (ӕ — 1)? 1 dx 9x² + 25 28. dx…

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Q: A) /16 /2 cos (167 x5 ) dx dy. Os xs 2 Og ys x*

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Q: Determine an equation for the sinusoidal function shown. 2. 12 21 Oy = 0.5 cos[0.5(x+ x)] + 1.5 Oy,…

A: The general equation of a sin function is given by; y=A cos(Bx+C)+D Where; A→Amplitude of the…

Q: sin(4x) cos (2x) dx = +(st)(9) (e) None sin sin (con (2x) Cos сов (10х) sin (2x) sin (10x) (e)( (Cо…

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Q: 1 dx 3 sin x+4 cos x+3

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Q: dy/dx+ytanx=cos^2(x)

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Q: 9. Find the value of k so that (2y sin x cos x - y+ 2y²e")dx - (x- sin? x+ kxye*)dy = becomes exact.…

A: To find the  value such that given equation is exact  differential equation.

Q: 4. tanx dy + y3 sin? x dx %3D 0, whe х %3D0, у%3D 1 о у?(3- сos 2x) %3D2 о у (сos 2x - 2x) %3D 1 о…

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Q: Following S sin(A) cos(B) dx = ; f sin(A + B) + sin(A – B), | what is the integral of f sin 5x cos…

A: We have to solve the following

Q: Example 1 Find dy/dx, if a. y=sin(x² + 3x + 1) b. y = cos² (2x-1)

A: I have stated the used formulas in the solution

Q: Question 4) A* dy Q4) A: Compute dx at x=1 where y = xcos2x O cos3 cos1 cos2 No one of the previous

A: Given y=xcos2x

Q: Question 9 z=cos y Using the substitutions the DE (3cos y – 5x) dx – 2x²tan y dy=0 reduces to dz= –…

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Q: d²s dt2 sin 3t + cos 3t |

A: In this question, we need to find s.

Q: 1. f(x) = In 5x 2. у %3D 103х Final Answer: Final Answer: 3. y = sin(e3×) 4. у %3 cos 2x (e**-1)…

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Q: A particle moves such that its velocity at time t is given by: v = 3t2i + 2 cos(4t)j + t2k If its…

A: given velocity of the particle at time 't' v=3t2i+2cos4tj+t2k initial position x(0)=2i

Q: 5. S sin(3x)dx (a) 3cos(3x)+ C (b) †cos(3x)+ C (c) – 3cos(3x)+ C

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Q: 7. Find (a) y = (x* + 5x) cos 5x %3D dx

A: We can find the derivative using product rule as below

Q: Suppose f (x) =cos( 2– 3x). Evaluate: )dx A - 3sin(3x – 2) +C ® -;sin(3x – 2) + C sin(3x – 2) + C D…

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Q: (3e + 10 cos(x)) dx ntegral =

A: Integration problem....