| (csc?x + 2 e2" dx = + 7* – 8) - O - cot x + 2e2a + - 8x + C - O tan x + e2a + 7° – 8x + C O - cot x + e2* + In(7) – 8x + C - - tan x + e2a + 7ª ln(7) – 8x + C
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- find dydx 15. y = (sec x + tan x)(sec x - tan x)16. y = x^2 cos x - 2x sin x - 2 cos x17. ƒ(x) = x^3 sin x cos x 18. g(x) = (2 - x) tan^2 xShow that cos x = x has a solution in the interval [0, 1]. Hint: Show that f(x) = x - cosx has a zero in [0, 1].Show that cos x = x has a solution in the interval [0, 1]. Hint: Show that f (x) = x − cos x has a zero in [0, 1].
- Find all solutions to cos(7x)−cos(x)=sin(4x)cos(7x)-cos(x)=sin(4x) on 0≤x<2π30≤x<2π3Suppose that the position of one particle at time t is given by x1 = 3 sin(t), y1 = 2 cos(t), 0 ≤ t ≤ 2? and the position of a second particle is given by x2 = −3 + cos(t), y2 = 1 + sin(t), 0 ≤ t ≤ 2?. (a) Graph the paths of both particles. How many points of intersection are there? __ points of intersection (b) Are any of these points of intersection collision points? That is, are the particles ever at the same place at the same time? If so, find the collision points. (Enter your answers as a comma-separated list of ordered pairs of the form (x, y). If there are no collision points, enter DNE.) (x, y) =__ (c) Describe what happens if the path of the second particle is given by x2 = 3 + cos(t), y2 = 1 + sin(t), 0 ≤ t ≤ 2?. The circle is centered at (x, y) = __ There are _ intersection point(s), and there are _ collision point(s).Evaluate Z 3π2πe1x sin x cos x dxusing Simpson rule with h =π20.
- Q4.) Show that y = (sin x +cos x) sec x = sec² xMy particular form solution looks like At^2+Bt+C+Dsin(4t)+Ecos(4t)+Fe^(2t), I know this is correct.I keep getting sin(4t)*(-16D-16E+4D)+cos(4t)*9-16E+16D+4E) which reduces to sin(4t)*(-12D-16E)+cos(4t)*9-12E+16D) which gets me D=(-4/3)E and E=-47/18 which is wrong, I even plugged it into an online diffeq solver which shows D=2/5 and E=-3/10 respectively. So I know there's something wrong with my application of method of undetermined coefficients.A rectangle is to be inscribed under the arch of the curve y = 4 cos (0.5x) from x = -π to x = π. What are the dimensions of the rectangle with largest area, and what is the largest area?