Question

How do you use the second fundamental theorem of Calculus to find the derivative of given `int cos 2t dt` from x to pi/4?

Answer 1

`1/4cos(x/2)-2cos2x`.

Explanation:

Using the substitution technique, together with appropriate changing of limits for the substituted variable, we get :

`int_x^(pi/4)cos2tdt=1/2int_(2x)^(pi/2)cosudu`, where `u=2t`

`=1/2[sinu]_(2x)^(x/2)`

`=1/2(sin(x/2)-sin2x)`.

We may now differentiate this result to obtain :

`d/dx[1/2(sin(x/2)-sin2x]=1/4cos(x/2)-2cos2x`.