We are given the function $f(x) = \cos(-2x)$ and are tasked with finding the constant of integration, $c$, for the integral $F(x) = \int f(x) \, dx$, given that $F(5) = -1.39$.

Step 1: Integrate $f(x) = \cos(-2x)$

We know that the integral of $\cos(kx)$ is $\frac{1}{k} \sin(kx)$, so we apply that here for $\cos(-2x)$:

$\int \cos(-2x) \, dx = \frac{1}{-2} \sin(-2x)$

Using the identity $\sin(-\theta) = -\sin(\theta)$, we get:

$\int \cos(-2x) \, dx = \frac{1}{-2} \cdot (-\sin(2x)) = \frac{1}{2} \sin(2x) + c$

Thus, the general form of $F(x)$ is:

$F(x) = \frac{1}{2} \sin(2x) + c$

Step 2: Use the given condition $F(5) = -1.39$

We now substitute $x = 5$ and set $F(5) = -1.39$ to solve for $c$:

$F(5) = \frac{1}{2} \sin(2 \cdot 5) + c = -1.39$

This simplifies to:

$\frac{1}{2} \sin(10) + c = -1.39$

Step 3: Calculate $\sin(10)$

Using a calculator to find $\sin(10)$ (in radians):

$\sin(10) \approx -0.5440211109$

Substitute this into the equation:

$\frac{1}{2} \cdot (-0.5440211109) + c = -1.39$

$-0.2720105555 + c = -1.39$

Step 4: Solve for $c$

Now, solve for $c$:

$c = -1.39 + 0.2720105555$

$c \approx -1.1179894445$

Thus, the constant of integration is approximately:

$c \approx -1.11798944$

Final Answer:

The constant of integration is $c \approx -1.11798944$.

Do you need further details or have any questions?

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Here are 5 related questions for further exploration:

1. What is the derivative of $F(x)$ and how does it relate to $f(x)$?
2. How would you compute the definite integral of $f(x)$ over a specific interval?
3. What happens if the given $F(5)$ condition changes to a different value?
4. How does the integration process change if the function was $\cos(3x)$ instead of $\cos(-2x)$?
5. What are the geometric interpretations of this integral in terms of the sine and cosine curves?

Tip: When integrating trigonometric functions, always check if angle identities or symmetry can simplify your calculations.