29. For |x|<1, prove that 24x 1-x 16x 8x 1 + 1-x1+x ....... 8x &r = 1+ (1- x) (1 +x)2 ' (1-x)2 ...

Q: sn'x プェ Sec'x-1 from [o, of 2.

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Q: 7. Find the area between f(x)=-x² + 4x and g(x) = x² - 6x +5 over the interval 0≤x≤1 Ei..

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Q: 2. Show that by using MEAN VALUE THEOREM. u-v n >0 1+v2 3 <tan- < and deduce that 25 ww 4

A: First we have to show that v-u1+v2&lt;tan-1v-tan-1u&lt;v-u1+u2 where 0&lt;u&lt;v. Then we will show…

Q: 15 Evaluate the integral: S 1+ xdx. Evaluate the integral: x2 + x³dx.

A: Given: I1=∫12151+xdx, I2=∫-13x2+x3dx We know that ∫xndx=xn+1n+1+c…

Q: Compute for the value of tanh (1 – i) Iny + 2.13i In 2 + 2.13i 2 In 5 - 4.25i 2 In v2 + 2.13i

A: To Compute: tanh-11-i Concept: 1. tanh-1z=12ln1+z-ln1-z2. lnz=lnr+iθ+2nπwhere,…

Q: 2. Assume that x > -1. Prove that (1+ x)" >1+ nx for n = 0, 1, 2, ....

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Q: f(x)=2sin(x)+4cos(x), then f'(x)=? f'(5)=?

A: Given  f(x)=2sin(x)+4cos(x)

Q: 73. Use the definition of |x\, the chain rule, and the fact that (In x) = - for x > 0 to prove that…

A: Given, dlnxdx=1x, x&gt;0 from definition of x x=x ,     x&gt;0 -x ,  x&lt;0  if x&gt;0 then d…

Q: ntiderivative. 7x x² -x-2 OB. In x + 1|-In x- 2| + c 14 .7 14 In x +1- x +1 In x-2 x- 2 3 3 14 7 OD.…

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Q: 5. Evaluate the integral. x+7 (x-6)(r-2)*a

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Q: If X is uniform in the interval [-3,3] compute X(i.e. E{X}) and o² (i.e.E{(X-X)²}).

A: Given that X ~ U[-3,3] to find mean and variance

Q: d Prove that (x") = n x"-1 dx

A:

Q: 1 1 The integral x²dxdy represents the 0, area under the curve y = x² between x = 0 and x = 1.…

A: Given integral is ∫01∫01x2dx dywe know that ∫ab∫cd f(x,y) dx dy represent areaonly when f(x,y)=1

Q: Find the linearization of y = eV1x at x = 36. (Use symbolic notation and fractions where needed.)…

A: We have to find linearisation

Q: 8. Evaluate the integral S xydx from x=3 to x=6 when x = 6cos0 and y = 2cose A. 9/7 B. 9/3 C. 5/3 D.…

A: Topic:- integration by substituting

Q: Prove that Jn (x) X good dx=1. n

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Q: In(1+2x) Find the Padé approximation R1,2(x) for f(r) %3D

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Q: 9 Evaluate the integral f 2x (x² + 1)° dx using the substitution = x² + 1. и — с 2.x (x? + 1)° dx +C

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Q: Evaluate the integral. 7 √49-x² √49-x²-22 49-x² 1 √²-3³ (x² + y² + 2²)1/2 dy dz dx =

A: Solution:-

Q: The evaluation of S V1 + tan2x dx is C In(V2) +1 In(vZ+1) C In(vZ-1) O-In(vZ+1)

A:

Q: x2-2 x+1 x2 --x-In Ix+11 + C C. + x- In Ix+ 11+C x2 -x + In Ix + 11 + C 2 +x + In Ix+ 11 + C O B

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Q: Find the general Selution yer r the equatiJen S/0x)-14 ()+ 0 By(x) = =D0 dlax

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Q: 1. Differentiate the following with respect to x: a. y= 5x - + 5Vx + 3p ex + In(x) - 7 (9)

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Q: 1 Evaluate the improper intergal | dx (x2 + 9)° using the residue theorem.

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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: хах 11. The integral f 4-х2 is equal to 1 A. In | | + C C.-글m14- x리 + C 4 x+2 2+x B. In |4 – x2|+C…

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Q: Let Z N(0, 1) and let y = [Z], where, for a real number x, [x] denotes the largest integer not…

A: Given,  Z ~ N(0,1) PDF of Z is given by, fZ(z) = 12πe-z22 ; z∈R Given, Y = Z So, Y = Z = 0if…

Q: e. y= (x² + 4)/x in the interval 1<x<4. 8+ 4ln4 - % s.u ans.

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Q: Solve for X(n) using any 2 out of 3 methods of the given X(z). 1 X(z) = (1– z--)(1– z-2)

A: Given z‐transform can be written as Xz=z3z-1z2-1=z3z+1z-12

Q: 4. Using the definition of sech x ,x 0 in terms of e , prove that 1+V1-x2 sech x = In ,0<x <1 Hence,…

A: 4) i)

Q: 6. so be it r = 3xi + yi – 2zk and f (r) : -- where |r| = r find all of grad(f) and div (-)

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Q: 8. Compute using the trig. substitution S x2 -dx (1–x²)1/2

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Q: Ôz Ôz If Z= tan"'(2) -= 0 show that : *-+ y ôx ôy

A: Let us consider function z=f(x,y) Where f is function of x,y. ∂z∂x represents partial derivative of…

Q: 7. a) Prove that for r > 0 * (E) + c. Vr2 – r²dx = vr² – x² + -1 sin 2 +C.

A: To prove: ∫r2-x2dx=x2r2-x2+r22sin-1xr+C

Q: 4. If the secant method is used on f(x) = x^5 +x^3 +3 and if x_(n-2) = 0 and x_(n-1) = 1, what is…

A: The given function is,

Q: 7. Consider the IVP y'=(2x-4)e" with y(5)=0. Verify that y = In(x² – 4x -4) is the solution to the…

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: 1. Evaluate the definite integral S ,2 C (9x² + 5)dx by the limit definition.

A:

Q: z/(z-3)2 dz ; C: |z+3| =3 slove by residue theorem

A: The given integral is,

Q: −x^2+x+6 has root(s) at x=

A: To find the roots of the given equation.

Q: 4. Show that In sin 6 d0 = – n 2 by applying the mean-value principle to In |1+ z| for |2| <r < 1,…

A: ∫0π2logsin(θ)dθ=12∫0πlogsinθdθ                         =12∫0π2logsin2θdθ…

Q: For Z= x²csc y,prove that Zxy is equal to Zyx

A: Given that,    z=x2csc y

Q: Q4 Legendra polmomial Pe (X) =2 + (3x+1) than Prove that (2u+1) Puck) =Pmnil (X) -P(X) n-1

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Q: x+1 Q1/ Prove that coth-1 x = 2 x-1

A: In this question we need to show coth-1x=12lnx+1x-1.

Q: – 30 - Evaluate the integral: -d.x x2 Vx2 + 16

A:

Q: 1 Evaluate the improper intergal dx (x² + 16)² using the residue theorem. 8.

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Q: In x + In(x – 1) = 10 In(x + 25) – In(x – 1) = 3 e2x – 2ex = 15 e2x – ex – 12 = Q %3D %3D -

A: i) ln x+ln(x-1)=10ii) ln(x+25)-ln(x-1)=3iii) e2x-2ex=15 Note:      Multiple part question are given,…

Q: Two real numbers X and Y are chosen at random in the interval (0, 1). Compute the probability that…

A: Selecting two points x and y from the interval (0,1) is identical to selecting a single point from…

Q: x+1 Q1/ Prove that coth-1x = x-

A: