Using Trapezoidal Rule find approximation of sin4xdx by divi

Using Trapezoidal Rule, find approximation of sin(4x)dx by dividing the domain into 3 intervals. Show that the following complex functions are differentiable everywhere and find the derivatives.

Solution

We have that a=1, b=3, n=3.

Therefore, x=313=23.

Divide interval [1,3] into n=3 subintervals of length x=2/3 with the following endpoints: a=1,5/3,7/3,3=b.

Prepare table as follows:

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w= 4ze^-2z

Use product rule

w\' = 4e^-2z-8ze^-2z

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w = e^5z/4

w\' = 5e^5z/4

x0	x1	x2	x3
1	1 2/3	2 2/3	3
f(1)	2f(5/3)	2f(8/3)	f(b)
-0.278412	0.1413361	0.017710424	-0.026714393	-0.146079869
Trapez rule value			-0.04869329