If In y = 3 sinh x show that dy (a) (1+r2)()² = 9y², dx (b) (1+ x²)- dy da? dy +x- – 9y = 0. da Hence, obtain the expansion of y in ascending power of r up to x*.

Q: Evaluate the integral. 3 1 3y2 cos xy dy dx x/3 5(1 - cos 3) 3(1 - cos 3) cos 3 - 1 (1 - cos 3)

A: Formulae used: ddxxn=nxn-1∫cos(ax)dx=sin(ax)a+C∫sin(x)dx=-cos(x)+Csin(0)=0cos(0)=1

Q: 10 Suppose that " fx) dx and Ax) dx = -5, and "g(x) dx = -1 and g(x) dx = 3. Compute the given…

A: Given,∫04f(x)dx=7∫02f(x)dx=-5∫04g(x)dx=-1∫02g(x)dx=3

Q: According to numerical methods, the integral of a function f(x) over the limits a and b can be…

A: Given data  Used Simpsons 1/3 rule  f(x)=7e^(-πx)(sin(x^2)) With a=0 and b=2 and number of intervals…

Q: Find General Solution on this Diffential E Qution by using (uC set) =e +Sin X

A: Given differential equation is y''+y'-2y=ex+sinxfind the general solution

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

A: Image is attached with detailed solution.

Q: In the integration of (cos y)^2 dy/(1 - sin y), what is the first step to solve the integral?…

A: ∫cosy21-siny dy

Q: 4. (a) (b) Approximate the integral f x² cos xdx with n = 6 using the composite trapezoid rule;…

A:

Q: Recall that 1, x >0 d (]xl) dx —1, х<0 x < d Use this result and the chain rule to find (sinx|) for…

A: Here, recall that  -π,-π2 is 3rd quadrant, where sin is -ve-π2,0 is 4th quadrant, where sin is…

Q: a)Evaluate y at position x=0.5  b)Evaluate the derivative dy/dx at position x=0.5. c)Using your…

A: For a given function of a single variable, its derivative represents at which rate the function is…

Q: mcory amu LAampIOS 45. Finding an upper limit of an iterated integral Solve for a: 1-x² ¢4-x²-y 4 dz…

A:

Q: Find the indefinite integral. |cos 2x ein 2* de sin 2x A) 2 +C ein 2x +C B) C) sin 2x e cos2x +C D)…

A:

Q: If xy In(3 sinh( 3x)), then x2y'= %3D Select one: Ox coth( 3x ) - In( 3 sinh(3x)) 09x coth( 3x ) - 3…

A:

Q: 2. The graph of y = cosh a is (a) concave up on its entire domain because y" = coshr > 0 for all r €…

A:

Q: In the integration of (cos y)^2 dy/(1 - sin y), what is the first step to solve the integral? Select…

A: Given integral.

Q: Which of the following is a left Riemann sum approximation of (x²) dx with n subintervals of equal…

A: Definition used-                  Left-hand Riemann sum -                   ∫abf(x)dx=…

Q: Use a second Taylor polynomial at x = 0 to estimate the areaunder the curve y = ln(1 + x^2) from x =…

A: Given that, y=ln1+x2 To estimate the area under the curve from x=0 to x=12, use the second Taylor…

Q: Find the integral of 12 sin^5 x cos^5 x dx if the lower limit is O and the upper limit is T/2.

A:

Q: In the integration of (cos y)^2 dy/(1 - sin y), what is the first step to solve the integral? a.…

A: Consider the integral: ∫cosy21-sinydy

Q: -) A) Find by using chain rule if f = ( , u = -1,x=-1 dx u+1 B) Evaluate the following integral:…

A: This question is based on chain rule differentiation and indefinite integral.

Q: 1. Use a Chain rule to find aw and ar aw , if w = x², x = sin(r²s) əs %3D

A: Given w = x2x = sin(r2s)

Q: 5 sin e cos ede 2. sin' 0+3 sin 6 -4

A: ∫5sinθcosθsin2θ+3sinθ-4dθ=5·∫sinθcosθsin2θ+3sinθ-4dθLet u=sin θ.=5·∫uu2+3u-4duBy partial…

Q: Show that for every 0 < p < 1 the integral 1 1 sin – dx xP is both improper and convergent.

A: Answer

Q: Question 1 00 is [ Select ] dx V2+5 The integral dx is [ Select ] The integral Vx2 + 5 [ Select ] dx…

A:

Q: 12.)Express lim E(3(x;,)² – 5x, + 2)Ax as a definite integral on the interval [0,1]. i=1

A: The given expression is: limn→∞ ∑i=1n 3xi2-5xi+2∆x    on interval [0, 1]   The definite integral  on…

Q: Eliminate the arbitrary constants 1. X = -Int, y= at + bt2 2. In y = C, sin ax + C, cos ax + 9x<-2…

A:

Q: Evaluate the integral. 3y- cos xy dy dx x/3 O 5(1 - cos 3) O 3(1 - cos 3) O cos 3 -1 (1 cos 3)

A: Given: ∫03∫x313y2 cos xy dy dx We have x3≤y≤1 and 0≤x≤3 This can be seen as: We can change the…

Q: At the point of discontinuity on(0, 00), the Fourier Sine integral converges to: OA None f(x+) -…

A: Here  At the point of discontinuity on  0,∞ the Fourier sine integral converges to

Q: Show that for every 0 < p < 1 the integral 1 1 sin = dx xP is both improper and convergent.

A:

Q: 3.12 Еxpress lim n00 n k3D cos 1+ as an integral. (Be sure to include the lower and upper limits|of…

A: Given: limn→∞ 2n∑k=1ncos1+2kn To convert this into the integral , use the following method

Q: 7. Use integration by parts to prove that for any positive integer n, п - 1 Scos | cos"x dx = 1…

A:

Q: Find the Fourier integral of the function f(x)= e* when x> 0 and f(-x) = f(x) for k > 0 cosix and…

A: Fourier integral, for even function

Q: or show that the integral Integrate the following diverges: Зе In sin x ii) dx COS X 2e

A:

Q: 25 3. In r If the triple integral fo So So (xe") dydr dz Find the upper limit value E.

A: We have to find the value of E.

Q: Jcos(2x)+ sin(0.5x) Using riemann compute for the integral of F(x) = tan(3x) sum over the interval…

A: We can solve this using Riemann sum

Q: An even piecewise continuous function f(x) on every finite interval of (-oo, 0) such that :/ (x)|…

A: The given statement is : An even piecewise continuous function f(x) on every finite interval of -∞,∞…

Q: - Prove using integrals Proper tos : e even sin()fc) dx=0 -b 9- For even functions.

A:

Q: |1) Use a sine expansion to compute the solution of ( ди д'и at ôx? u(0,t)=u(1,t) = 0,t > 0, u(x,0)…

A: According to the given information, it is required to compute the solution of the given equations.

Q: Evaluate the integral: sinh 3x csch 3x dx coth 3x "+C 3 coth X +C 3 - coth x +C - 3 - coth 3x +C 3

A:

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

A: Use the following formulae, to evaluate the given complex integral. Let z0 be a pole of order n,…

Q: Evaluate if the integral converges 1. 1. dx (x-1)7 , sin x dx 3. dx 2.

A:

Q: Find the Wronskian for y, = sin 3x , and y2 = cos 3x.

A: The Wronskian, W, of two functions, f and g is calculated as follows. W=fgf'g' , where, f' , g'are…

Q: Find Z in (1-j5) 2= tanh cos lat5 express in lagarithm Form

A:

Q: (a) Explain why | dr is divergent. (b) Find all value(s) of t > 0 for which the integral / 1 dx is…

A:

Q: Evaluate the indefinite integral as an infinite series. arctan(x5) dx Σ( |)•c n = 0

A:

Q: (a) (6) Decide if the following improper integrals are convergent or divergent using Basic…

A: To decide the convergence of the improper integrals.

Q: 1. Prove: a. csch x In +c %3D b. sech x = 2 Arctan(e*) + c %3D

A: 1. (a). To prove: ∫cschx=lnex-1ex+1+C Proof: We know that, sinhx=ex-e-x2 The left hand side can be…

Q: S az lnz (In z + 1)dz z In

A: ∫azlnzlnz+1 dz Put, z lnz=u⇒1·lnz+1z·z=dudz⇒lnz+1=dudz⇒lnz+1 dz=du

Q: Only piecewise continuous functionsf (x) on every finite interval (0, 0) such that 00 | S(x)| dxis…

A: According to the condition of convergence for Fourier integral,  Let f and f' be piecewise…

Q: Evaluate Poisson Integral: In(1-2a cos(x) + a²) dx ,a.) [a| 1,

A: To evaluate Poisson's integral: ∫0πln1-2αcos(x)+α2dx  a)α&lt;1 b)α&gt;1

Q: If In |(In x)" dx, show that In = x(In x)" – n.Ip-1- Hence find nx)* dx.

A: I am going to prove the given identity by using integration by parts and then use this identity to…