d/dx integral from o to x of function cos(2*pi*x) du is

first i do the integral and i find the derivative right.

by the fundamental theorem of calculus, if there is an integral from o to x, don't i just plug the x in the function.

the integral of the problem is cos*2*pi*) is just cos(2*pi*x) right or is that wrong. then i find the derivative.

<<the integral of the problem is cos*2*pi*x) is just cos(2*pi*x) right or is that wrong.>>
Wrong. The integral of that function is
[1(2 pi)]*sin(2*pi*x)

To find the derivative of the integral from 0 to x of the function cos(2*pi*x) du, you can apply the Fundamental Theorem of Calculus.

The first step is to evaluate the integral. In this case, you are integrating the function cos(2*pi*x) with respect to du over the interval from 0 to x. Integrating cos(2*pi*x) with respect to du will simply give you cos(2*pi*x) * u.

So, the integral from 0 to x of cos(2*pi*x) du is equal to cos(2*pi*x) * u evaluated from 0 to x.

Now, we can substitute the upper limit (x) and the lower limit (0) into the expression.

At x, the result is cos(2*pi*x) * x, and at 0, the result is cos(2*pi*0) * 0.

Since cos(0) = 1 and multiplying by 0 gives 0, the expression evaluates to:

cos(2*pi*x) * x - 0 * 0
= cos(2*pi*x) * x

Next, you want to find the derivative of this expression with respect to x.

So, taking the derivative with respect to x, you apply the product rule:

d/dx [cos(2*pi*x) * x] = (d/dx [cos(2*pi*x)]) * x + cos(2*pi*x) * (d/dx [x])

The derivative of cos(2*pi*x) with respect to x is -sin(2*pi*x) * (2*pi), and the derivative of x with respect to x is 1.

Therefore, the derivative of the integral from 0 to x of cos(2*pi*x) du is:

-sin(2*pi*x) * (2*pi) * x + cos(2*pi*x) * 1
= -2*pi*x*sin(2*pi*x) + cos(2*pi*x)