We are tasked with evaluating the integral $\int \cos(8x) \, dx$, using the substitution $u = 8x$.

Steps to Solve:

1. Substitute $u = 8x$: $\frac{du}{dx} = 8 \implies dx = \frac{du}{8}$

2. Rewrite the integral: Substitute $u$ and $dx$ into the integral: $\int \cos(8x) \, dx = \int \cos(u) \cdot \frac{du}{8}$

3. Factor out the constant $\frac{1}{8}$: $\int \cos(8x) \, dx = \frac{1}{8} \int \cos(u) \, du$

4. Integrate $\cos(u)$: The integral of $\cos(u)$ is $\sin(u)$: $\frac{1}{8} \int \cos(u) \, du = \frac{1}{8} \sin(u) + C$

5. Substitute back $u = 8x$: Replace $u$ with $8x$: $\int \cos(8x) \, dx = \frac{1}{8} \sin(8x) + C$

Final Answer:

$\int \cos(8x) \, dx = \frac{1}{8} \sin(8x) + C$

Would you like further explanation or examples related to this topic?

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Here are 5 related questions for practice:

1. Evaluate $\int \sin(5x) \, dx$ using a similar substitution.
2. What is the integral of $\cos(ax + b)$ with respect to $x$?
3. Solve $\int \cos^2(4x) \, dx$ using trigonometric identities.
4. Find the derivative of $\frac{1}{8} \sin(8x) + C$ to verify the solution.
5. Evaluate $\int e^{3x} \cos(5x) \, dx$ using integration by parts.

Tip: Substitution is often helpful when dealing with composite functions, especially when the inner function's derivative appears in the integrand