integral cosine of two times x minus seven d x equals

To evaluate the integral \(\int \cos(2x - 7) \, dx\), we can use a substitution method. Here's the step-by-step process:

### Step 1: Substitution
Let \( u = 2x - 7 \).

Now, differentiate \( u \) with respect to \( x \):
\[
\frac{du}{dx} = 2 \quad \text{or} \quad du = 2 \, dx
\]
This means:
\[
dx = \frac{du}{2}
\]

### Step 2: Substitute into the integral
Substitute \( u \) and \( dx \) into the integral:
\[
\int \cos(2x - 7) \, dx = \int \cos(u) \cdot \frac{du}{2}
\]
Simplify the integral:
\[
\frac{1}{2} \int \cos(u) \, du
\]

### Step 3: Integrate
The integral of \( \cos(u) \) with respect to \( u \) is \( \sin(u) \):
\[
\frac{1}{2} \int \cos(u) \, du = \frac{1}{2} \sin(u) + C
\]

### Step 4: Substitute back the original variable
Replace \( u \) with \( 2x - 7 \):
\[
\frac{1}{2} \sin(2x - 7) + C
\]

### Final Answer
\[
\boxed{\int \cos(2x - 7) \, dx = \frac{1}{2} \sin(2x - 7) + C}
\]

Would you like more details on this process, or do you have any questions?

Here are 5 related questions to consider:
1. How would you solve the integral of \(\sin(2x - 7)\)?
2. What if the integral was \(\int \cos^2(2x - 7) \, dx\)?
3. How does the substitution method help in solving trigonometric integrals?
4. Can this integral be solved using integration by parts?
5. How would the integral change if it was \(\int \cos(3x - 7) \, dx\)?

**Tip:** Always check if a simple substitution can simplify the integral before trying more complex techniques like integration by parts.

Explore how to evaluate the integral of cosine 2x - 7 using the substitution method. This detailed guide provides a step-by-step solution, including integration techniques and trigonometric formulas. Ideal for advanced high school students and calculus enthusiasts.

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