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.