\displaystyle \int {\cos \left( 2\,x-7 \right)} \, \textrm{d}{x} =

To solve the integral \(\int \cos(2x - 7) \, \mathrm{d}x\), we can use the following steps:

1. **Substitution**: Let \( u = 2x - 7 \). Then, differentiate \( u \) with respect to \( x \):
   \[
   \frac{du}{dx} = 2 \implies du = 2 \, dx
   \]
   This implies that \( dx = \frac{du}{2} \).

2. **Substitute**: Replace \( 2x - 7 \) with \( u \) and \( dx \) with \( \frac{du}{2} \):
   \[
   \int \cos(2x - 7) \, \mathrm{d}x = \int \cos(u) \cdot \frac{du}{2} = \frac{1}{2} \int \cos(u) \, du
   \]

3. **Integrate**: The integral of \( \cos(u) \) with respect to \( u \) is \( \sin(u) \):
   \[
   \frac{1}{2} \int \cos(u) \, du = \frac{1}{2} \sin(u) + C
   \]
   where \( C \) is the constant of integration.

4. **Back-substitute**: Replace \( u \) with \( 2x - 7 \) to get the final answer:
   \[
   \frac{1}{2} \sin(2x - 7) + C
   \]

So the final result is:
\[
\int \cos(2x - 7) \, \mathrm{d}x = \frac{1}{2} \sin(2x - 7) + C
\]

Would you like any further details or explanations? Here are some related questions:

1. What is the integral of \(\sin(2x - 7)\)?
2. How do we integrate \(\cos^2(2x - 7)\)?
3. Can you explain the substitution method in more detail?
4. What happens if the coefficient of \(x\) in the cosine function is not 2?
5. How would you differentiate \(\sin(2x - 7)\)?

**Tip:** When performing substitution in integrals, always remember to adjust the differential \(dx\) accordingly to avoid mistakes.

Learn how to integrate cos(2x - 7) step-by-step. This problem involves the substitution method in integration, focusing on trigonometric functions. Suitable for advanced high school students and beyond.

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