'r+ 1, 1피<, f(x) = 0, |a|> 7.

Q: Find a function f € span(1, cos x, sin x) which minimizes the integral 2πT √²* |x +1 − f(x)| ²dx.

A: Given a function                     f∈span1, cosx, sinx

Q: 1-X 1.S.S C14 X) dY dX (1-x) dy dx 1+X

A:

Q: 2 S (x) + x³)dx = ? -1

A:

Q: find integral x3 3) S dx /1-x²

A: We have to find the indefinite integral

Q: 2 3x 2. Evaluate f x'e* using integration by parts.

A:

Q: ential equati x²y dy = C

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Q: Solve ODE for general solution: (2ysin(x)cos(x)-y+2y^(2)e^(xy^2))dx=(x-sin^2(x)-4xye^(xy^2)dy

A: Hence the given ode is exact. Obtain the general solution as follows,

Q: 10. f (x) = x² + 4x + 4 Opens up/down:. Vertex: Axis of symmetry: Solution(s): y-int: Domain: Range:

A:

Q: 1) = sinx + ytanx dy dx 2) x = y + x sin? 3) Prove that (2xy – 9x2)dx + (2y + x² + 1)dy = 0 is…

A: 1   dydx=sinx+ytanx        the equation can be written as standard form        dydx-ytanx=sinx…

Q: Q4/ 1- Evaluate: f/2fy sinx dy dx ди 2- If u = x² + ey² ,x= sin 2t , y = cost² find at

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Q: 1 Evaluate the improper integral dx. (x – 3)3/2' 3 -

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Q: Determine | An (x) An+1(x) dx, from the relation: tn .1 Σ (2n + 1)! tn+1 A, (x) An+1(x) dx n=0 n -1

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Q: If f(x) = A B (C) tanx S 0 sec ³x 1+e tanx 1+tan²x 1+ e tanx 1+12 sec²x 1+e tanx In (coor) 1+ e'…

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Q: integral from 1 to 2 of (e1/x^3 ) / x4

A: To evaluate

Q: y 2sin x-cos 2.x for 0° sxs360

A: Consider the function, y=2sinx-cos2x Differentiate with respect of x,…

Q: Find the solution and the interval of existence for: (1-x2)dy/dx + 2y = (1-x)2

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Q: y = x - 2 sin(x), 0 sxs 27 aximum value inimum value

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Q: 1. Obtain the General Solution. Use reduction to exact. Simplify your fin y(y + 2x- 2) dx - 2(x +…

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Q: 8. Find the length of the curve y = e on the interval 0, ln 2| %3D

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Q: Find the general solution using Reduction of Order: y''+y'=cos 3x

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Q: Find the solution and the interval of existence for  (1-x2)dy/dx +2y = (1-x)2

A: The objective is to find the solution and existence for  1-x2dydx+2y=1-x2dydx+21-x2y=1-x1+x The…

Q: fy = f(r), how do we define the differential dy

A:

Q: 12. Show that F(x) = sinx – x cos x+2 is an antiderivative of f(x) = x sinx.

A: Given, F(x)=sinx-xcosx+2 To show that the above function is an antiderivative of f(x)=xsins.

Q: From the differential equati on by eliminating the arbitrary function ffrom xyz = f (x?+y'+z°).

A:

Q: π/2 g(x) = 2x + 4 sin² x g(t) dt. - n 0

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Q: In integrating the function f(3x + 5)(cos) dv using integration by parts, what would you choose for…

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Q: Prove: ex. a. ſ csch x = In b. S sech x = 2 Arctan(e*) + c

A: a).  ∫cschx dx=∫2ex-e-x dx=∫2ex-1ex dx=∫2exex2-1 dx Put, ex=t⇒ex=dtdx⇒ex dx=dt Therefore, ∫cschx…

Q: R(x) = F/2 π dx

A: We have to evaluate the integral: R→x=∫0π/2sin2x3x+1, lnx2-1,xx2+1dx   .....(1)

Q: 7 Integrate by using the Archimedes-Riemann Theorem | (2.02 + x)dx

A:

Q: dx J x2-4x+5

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

Q: The linearization of ex at x = 0 Derive the linear approximation ex = 1 + x at x = 0.

A: Given, ex ,x=0 We know that, The formula for the linearization is, Lx=fa+x-af'a

Q: y" +y' = cos 3x

A:

Q: 1 The approximation of dx by the composite Trapezoidal Rule with n= 2 is +4 O X-

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Q: The approximation of I = f(x-3)** dx by composite Trapezoidal rule with n=4 is:

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Q: Solve integration by (udv) T/2 DS Š X cos(x) dx 2) { X cos (ux) dx 7/2

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Q: approximation of I = [" f(x)dx using composite Simpson's rule with n=4 is:

A: In this question, we evaluate the definite integral by using the composite Simpson's rule for n=4.

Q: Q.4 Using the method of substitution the integral fxer-3 dx

A:

Q: se N.P.M. to estimate the root of -X = e -X for an initial of Xo =0

A: Here f(x)=e-x-x,  x0=0. We estimate the root of f(x)=0 by using N.R.M.

Q: Sx". .n-1 + n[x²- x" sin x dx = -x cos x + n cos x dx

A:

Q: S sin t dt 0. lim x→0 2 x4

A: Given,

Q: Q2: - If y = 3sec3t and x = etan3t prove that: - dy (In3)2 + dx y 2logy x %3D

A: we have ,  y=3sec3t and x=etan3t we have to prove that , dydx=(ln3)2·yx·logyx…

Q: 4. Prove 1 < | V1+ a dx <2 0,

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Q: 2 f(x)cos(n@,x)dx = %3D 27 cos(nx)ch. sin(nx) cos(nx) (2x) -sin(nx) + 2 11 f(x)sin(n@,x)dx =- [x…

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Q: Limit 0 to 2 √x dx

A: We have to find the definite integration: ∫02xdx We know the formula of integration, ∫xndx=xn+1n+1+C…

Q: 4. Prove that ах x" ea" dx xn- п-1 ‚ax dx а а

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Q: 39. Use Leibniz's Rule to find the value of x that maximizes the value of the integral px+3 1(5 – t)…

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Q: Test for homogeneity

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Q: .3 cos(t) dt F(x)%3 cos(x). *

A: We are going to solve this question by the method of leibniz rule of differentiation under integral…

Q: prove using the IVT. e^x + ln x = 0 has a solution.

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Q: 1. What should be A for the path independent of (2,4) xy + A dx + 2Ay – x y? fip- (1,2) Moreover…

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