Prove that Σ n=1 sin ne n - 6.² < sin x X - dx = 1.8519... for any positive integer m and any 0 in [0, π]. Show moreover that the constant on the right-hand side cannot be replaced by any smaller number.

Q: 29. Show that sin x rif0<1 <2m.

A: Given a function sin x and 0&lt;x&lt;2π To show sin x&lt;x for all values of x such that…

Q: 4. Find two possible non-linear functions & and such that goh= √sinx-x²

A: To find the non linear function by the given composition.

Q: If n is an odd natural number prove that I =, xsinnx dx = sinx 2.

A: Consider the given: It is given that n is an odd natural number, Take n = 1,Substituting n=1 and…

Q: s if 0 < x <277 -cos 2xsin x

A:

Q: I-Shinx dx Ig-/sin'xdx 2-x2 dx

A: We have to evaluate the above integrals. NOTE - According to our guidelines we have to answer only…

Q: 25. The height of a wave can be represented by the function y = 3 cos ((2 - 6)) + 4, with time in z…

A: The equation represents a periodic function. The height is represented by and time in hours is…

Q: If I, = [x^tan"x dx, Prove that (n+1), + (n-1),,2 ° п 1 %3D 2 n

A:

Q: Show thatI cos ve (-1/4s)

A:

Q: 18 a Show that / (1 + sin 2zx)dx = n, for all positive integers n. b Sketch y = 1 + sin 27x, and…

A:

Q: 1. Show that cosh 4x = 8cosh“x - 8cosh? x + 1 .

A: Given problem:- Show that cosh(4x) = 8cosh4x - 8cosh2x + 1

Q: Evaluate p* sin 0 de dr.

A:

Q: EVALUATE -dx + 144 ,2 O x – 12tan-1 12 none 1 -sin' 12 x – -1 +c 12 O x – tan-1 12

A: We have to evaluate  ∫x2x2 + 144 dx. We know,  ∫1x2+a2dx = 1atan-1xa + C      (where C is constant…

Q: 2c. Show that * (5 - ) 4 sin z - cos 2r + 3 -8 cos for every r.

A: To prove: 4sinx-cos2x+3 = 8cos4π4-x2 We know that cosx-y  = cosxcosy+sinxsiny and cos2x= 1-2sin2x

Q: 77. Find - Ans. (sin t – cos t + e-t) (8+ 1)(82 + 1) |

A: Topic:- inverse Laplace transform

Q: 9. (1+ sin 0) de

A: To evaluate the given integral

Q: Find sin a. (-/5 ,2) [ ?] 1

A:

Q: Q/ Evaluate the following functions 1.In(3-16) 3. cos ¹ (1+i 3) 2 (3|0.52) 2. e ‚4. tanh¯¹ (2-i5)

A: According to the given data, evaluate the functions:

Q: Use the addition formulas for sinhx and coshx to prove that tanh(u+v)= tanhu+tanhv/ 1+tanhutanhv

A:

Q: 2 4<3 tan( 37x)

A: Solve 2tan37x-4&lt;3

Q: d) √3 cos x + sin x = 0 [-2π, 0]

A: By converting the given equation in single trigonometric function,, 99, 254); position: relative;…

Q: sin mx = m sin x 1+m 1-m 3 2 2 2 sin x

A:

Q: Prove that fx" cos(2x)dx=sin(2x)-[x n sin(2x)-x-¹ sin(2x)dx. Do not use Table Entries. H

A:

Q: Show that cosh x> 1 + x² for all x.

A: comparison

Q: Q2) a) Find f xein(x²+5) cos(x? + 5) dx

A:

Q: Determine 1-/[e-tt sin(2t)] 2- 23+3s)*

A: Given

Q: Show that 1 − cos x = x2/2 with an error less than 0.003 for |x| &lt; 1/2.

A: Given: 1-cos x =x22;    |error|&lt;0.003Formula:  The…