x[n] - 4 Cos(0.3 7in)+ 7 Sin(0.25 Tin)

Find the n th derivative of: (1)cos4x

If x(n) = [1,2,0,3, -2,4,7,5] evaluate the following i) X (0) ii) X (4) iii) ∑ 𝑋(𝐾) 7 𝑘=0 iv) ∑ |𝑋(𝐾)| 7 𝑘=0 2

∫e sin(x) cos(x)dx.

sinn x dx = − 1n sinn−1 x · cos x + n − 1n∫sinn−2 x dx. (6.4)If we take dv = sin x dx, then we have v = − cos x and we may integrate by parts withu = sinn−1 x, du = (n − 1) sinn−2 x · cos x.Using the fact that sin2 x + cos2 x = 1, one may thus conclude that∫sinn x dx = − sinn−1 x · cos x + (n − 1)∫sinn−2 x · cos2 x dx= − sinn−1 x · cos x + (n − 1)∫sinn−2 x · (1 − sin2 x) dx= − sinn−1 x · cos x + (n − 1)∫sinn−2 x dx + (1 − n)∫sinn x dx.Here, the rightmost integral coincides with the original integral on the left. Once we nowrearrange terms, we end up with n copies of the integral and equation (6.4) follows. Example 6.11 We use a reduction formula to compute the integral I3 in the case thatIn =∫xne2x dx.If we take u = xn and dv = e2x dx, then du = nxn−1 dx and v = 12 e2x, so one hasIn = 12 xne2x − n2∫xn−1e2x dx = 12 xne2x − n2 · In−1. (6.5)We now apply the last formula repeatedly to determine I3. According to the formula,I3 = 12 x3e2x − 32 · I2 = 12 x3e2x − 32 ·[12 x2e2x − I1]= 12 x3e2x − 32 ·[12 x2e2x − 12 xe2x + 12 · I0]= 12 x3e2x − 34 x2e2x + 34 xe2x − 34∫e2x dx= 12 x3e2x − 34 x2e2x + 34 xe2x − 38 e2x + C. 

Use trigonometric identities and algebraic methods, as necessary, to solve the following trigonometric equation. Please identify all possible solutions by including all answers in [0,2π)[0,2𝜋) and indicating the remaining answers by using n to represent any integer. Round your answer to four decimal places, if necessary. If there is no solution, indicate "No Solution."cos2(3x)=sin2(3x)