9/ Evalaale he fillow inlag rad S sin'z asx dy x CoSx

Q: 1 Show that L-1{ = (sin hat sin at) s4+4a4, 2a²

A: It is given that, L-1ss4+4a4=12a2sinhat sinat. A partial fraction decomposition of ss4+4a4:…

Q: 3) Prove that c) cosh2x = 1 + 2 sinh? x

A: In the question we have to prove the given function.

Q: Differentiate: y = sinh° (sin 5x) Oy' = 15 cos 5x sinh (sin 5x) cosh(sin 5x) Oy' = –15 sin 5x sinh?…

A: dsinh (ax)dx=a cosh (ax) dsin(ax)dx=a.cos(ax)

Q: 4) a²y=+52y =14 yat (2,1) Find equátion of tangent?

A: Equation of tangent  y-y1=m(x-x1) (x1,y1)are the points at which equation of tangent is to be…

Q: find /x – x² - dx xV4x – x² dx 9. 10. х dx dx 11. 12. xV7 – x² xV7 + x² Vx² (4 x² 4 dx 13. 14. х sin…

A:

Q: Suppose tan u = %¾ (sm u>0) and tanuo12/5 Ccasx cO]. Ead he exsee velos OR the followly trgeronetnc…

A:

Q: リ=!-Cos2x 1+(os 2x prove y = 2 tanx .secx

A:

Q: Q3. Show that дхду for дудх f = x² siny – 2y³ cos2x (

A:

Q: b,c,d

A:

Q: 7 show that: sech x dx = Sin (tankx). √1-u² multiply note: * & Sim'u = dx * you Cam سال dx sechx by…

A: Differentiate rhs

Q: Calculate for w-z sin(xy+).

A: The given function is w=zsinxy4+9z

Q: If y = √√1-x² sin¹¹x, prove that (1-x²)+xy=1-x². dx

A:

Q: Find Domain and Range бяц tue Ⓒsin (x+3) +2y x²+2y 3x + y² 3) 3x + Sin (3x) +2y+Z² Ⓒ x+2y cos2x…

A:

Q: 1. If Y= 4 In (In (In(sec(x)) then Ý=[4 (tan)/{ In ((In (sec)) In (sex)}]

A:

Q: b) Find if Scosy Vt3 + 5 dt + tany = x -sinx b) Find if S* Vt2 + 5 dt + siny = x – x°cosy dx2

A: Solve by differentiation

Q: If Z= sinh () show that: *=+y==0

A: GivenZ=sinhyx

Q: zy (Smxa CosX) Axー Sinhy

A:

Q: If f(x,y,z)=In(x²y+sin² 12

A:

Q: (1-1/2 x^2)cos x 3. y = 2cosxsin2x-sinxcos2x 4. y= x^3 - x^2cosx + 2xsinx + 2

A: Differentiation: 1.y=sinx2+cosxdydx=(2+cos x).cosx-sinx(-sinx)(2+cos…

Q: f'(t) = sin 8t - cos -/-

A: You are correct

Q: (A) Prove that: cosh x + sinh x = e*

A:

Q: Show that bothy= sinx and y = cos x satisfy y" = -y

A: Derivatives can be defined as rate of change of function with respect to an independent variable. It…

Q: Find ds/dt if 3t + 3s sin(3t) + 3cos(t) %3D

A:

Q: Evaluate the integral. sin (8t + 16) dt cos (8t + 16) sin (8t + 16) dt = cos (8t + 16)

A: Solved below

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: The integral S sine de 1+ cose sec²0 de 1+ tan cose de 1-sine sece tane de sece-1 S x dx x²+√√x²-1-1…

A: dsecθdθ= sec θ.tan θ

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

A:

Q: Compute Sinh Z c (az - z where C 121-1

A:

Q: 2. Show that the LT of sin(wt) is L sin(wt)] (s) where w is a positive constant.

A:

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: 4. Use definition (not tables) of L {f(t)}= [ e"f(t)dt to find L{sin(2t)} . %3D

A:

Q: Which of the following is true? J sin(3x)sin(4x)dx = cos(x) – cos(7x) dx 2 ® Jsin(3x)cos(4x)dx = […

A:

Q: Evaluate the integral. sin (8t + 9) -dt. cos (8t +9) sin (8t + 9) dt%3D cos (8t+ 9)

A:

Q: find do diy y = (cos(x))*os)

A: Given y=(cos(x))^(cosx) Now differentiate y with respect to x

Q: |1- x 2 arctan + arcsin(2x – 1) = Hint using f Odx = C

A:

Q: 9 If f(0) = cos 01 ·cos 02·cos 03 ....cos 0n %3D Show, {tan 0 + tan 02 + tan 03 + ... + tan 0,}=- f…

A: First note the results derivative of log x is 1x and derivative of cos(x) is -sin(x). The proof of…

Q: if t(い) = ta Cosy V = X siny Determine JR ļ JR to つメ

A:

Q: 14. Use a suitable change of variables to determine the indefinite integral. fsin²xcos³ xdx (Hint:…

A:

Q: b) Find if e Vt2 +5 dt + siny = x- xcosy rsinx ycscx

A: According to Leibniz' rule:

Q: 2 = sin5t constitute a fundamenta L

A: Y1=cos5t and y2=sin5t  do not a fundamental set of solutions of differential equations y"+16y=0

Q: Evaluate the integral. sin (6t + 17) -dt CoS 2 (6t + 17) sin (6t + 17) dt%3D cos 2 (6t + 17)

A:

Q: B) Prove that: cosh (x + y) = coshx coshy + sinhx sinhy %3D

A: We know that cosh(θ) = eθ+e-θ2sinh(θ) = eθ-e-θ2

Q: Show that Sx cos*x dx = (x² + x sin 2x +cos 2x) + c

A: Given: ∫xcos2x dx=14x2+xsin2x+12cos2x+C.

Q: 4. Use Secant method with p =1, p =1.2 to find the root of f(x)= Inx – cosx that is accurate to…

A: Given, fx=ln x-cos x; p0=1; p1=1.2; As it is known that the secant method is given as,…

Q: find tan x dx x² x² tan¬' x dx 19. 20. sin 3x cos 2x dx 22. sin 2x cos 3x dx S sin sin 8 sin 4t sin…

A:

Q: 3 Differentiate: y = In cosh° (cos 2x)

A: As we know that: The chain rule: fgx'=f'gx·g'x The Power law: ddxxn=nxn-1 From the given problem:…

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

A: Solving the problem by finding the integrand.

Q: Prove that for all x € [1, 00), – sinh (- |cosh'(x)|) = V -1. %3D Ouest!

A: Proving of function  sinh(cos-1(x))

Q: 4. Use definition (not tables) of L{f(t)}= [ e"f(t)dt to find L{sin(2t)} .

A:

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.