Find the following indefinite integral.
Integral from nothing to nothing left parenthesis cosine x plus 8 sine x right parenthesis dx

To find the indefinite integral of the expression \(\cos(x) + 8\sin(x)\) with respect to \(x\), we proceed as follows:

\[
\int (\cos(x) + 8\sin(x)) \, dx
\]

We can split the integral into two separate integrals:

\[
\int \cos(x) \, dx + \int 8\sin(x) \, dx
\]

Now, we integrate each term separately:

1. The integral of \(\cos(x)\) with respect to \(x\) is \(\sin(x)\).
2. The integral of \(\sin(x)\) with respect to \(x\) is \(-\cos(x)\), and multiplying by 8 gives \(-8\cos(x)\).

Putting it all together, we have:

\[
\sin(x) - 8\cos(x) + C
\]

where \(C\) is the constant of integration.

So the final answer is:

\[
\int (\cos(x) + 8\sin(x)) \, dx = \sin(x) - 8\cos(x) + C
\]

Would you like any further details or have any questions about the steps involved?

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Here are 5 related questions to consider:
1. How do you compute the definite integral of \(\cos(x)\) from 0 to \(\pi/2\)?
2. What is the indefinite integral of \(\cos^2(x)\) with respect to \(x\)?
3. How does the integral of \(\sin(x)\) change if the limits are from \(-\pi\) to \(\pi\)?
4. What are the integrals of trigonometric functions like \(\tan(x)\) or \(\sec(x)\)?
5. Can you find the integral of \(\sin(x)\cos(x)\)?

**Tip:** When integrating, always check if the function can be simplified or split into parts to make the integration process easier.

Learn how to find the indefinite integral of cos(x) + 8sin(x) with a step-by-step solution. This problem covers integration of trigonometric functions and is suitable for high school students.

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