Consider the function  f ( x ) = cos ( − 2 x ) , and suppose that  F ( x ) = ∫ f ( x )   d x .  If  F ( 5 ) = − 1.39 , then find the constant of integration,  c Enter as many decimal places as your calculator allows (8 to 10). Your answer must be within  ± 0.005  of the correct answer to be considered correct

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.

Consider the function f(x) = cos(-2x), and suppose that F(x) = ∫ f(x) dx. If F(5) = −1.39, then find the constant of integration, c. Enter as many decimal places as your calculator allows (8 to 10). Your answer must be within ±0.005 of the correct answer to be considered correct.

Solve the calculus problem of finding the constant of integration for the function F(x) = ∫ cos(-2x) dx, given F(5) = -1.39. This problem covers key concepts in trigonometric integration and the Fundamental Theorem of Calculus. Suitable for students in Grades 11-12.

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