Question

What is C?

`f'(x)=2sec(2x)tan(2x)`. `f(pi/2)=.5`. Find C

Answer 1

`C = 1.5`.

Explanation:

C is often used to represent the constant of integration. Since we are given the value of `x` and `y` on `f(x)`, we will be able to solve for `C` after integrating.

I'm going to start by rewriting as a differential equation.

`dy/dx = 2sec(2x)tan(2x)`

`dy= 2sec(2x)tan(2x)dx`

`int dy = int 2sec(2x)tan(2x)dx`

If we let `u = 2x`, then `du = 2dx` and `dx = (du)/2`.

`int dy = 1/2 int2secutanu du`

`y = int secutanu du`

This is a known integral

`y = secu + C`

`y = sec(2x) + C`

We can now solve for `C`.

`0.5 = sec(2(pi/2)) + C`

`0.5 = 1/cos(pi) + C`

`0.5 = -1 + C`

`1.5 = C`

Hopefully this helps!