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Let's evaluate the given indefinite integrals step by step.

### a) \(\int 10 \sin x \, dx\)
Given: \( \int 10 \sin x \, dx = 10 \cos(x) + C \)

This is correct because:
\[
\int \sin x \, dx = -\cos x
\]
So, multiplying by 10:
\[
\int 10 \sin x \, dx = 10(-\cos x) = -10 \cos x + C
\]

Thus, the correct expression is:
\[
-10 \cos x + C
\]

### b) \(\int 8 \sec^2 s \, ds\)
We know that:
\[
\int \sec^2 s \, ds = \tan s
\]
So,
\[
\int 8 \sec^2 s \, ds = 8 \tan s + C
\]

### c) \(\int (15 \sec t \tan t + 17 \csc t \cot t) \, dt\)
We can break this integral into two parts:
1. \(\int 15 \sec t \tan t \, dt\)
2. \(\int 17 \csc t \cot t \, dt\)

- The integral of \(\sec t \tan t\) is \(\sec t\), so:
  \[
  \int 15 \sec t \tan t \, dt = 15 \sec t
  \]
  
- The integral of \(\csc t \cot t\) is \(-\csc t\), so:
  \[
  \int 17 \csc t \cot t \, dt = 17 (-\csc t) = -17 \csc t
  \]

Therefore, the combined result is:
\[
15 \sec t - 17 \csc t + C
\]

### Final Answers:
a) \(-10 \cos x + C\)

b) \(8 \tan s + C\)

c) \(15 \sec t - 17 \csc t + C\)

Would you like further details or explanations on any step?

Here are 5 related questions you might want to explore:
1. How can you verify an indefinite integral?
2. What is the difference between definite and indefinite integrals?
3. How do you handle integrals of trigonometric functions in general?
4. What are the integration rules for inverse trigonometric functions?
5. How does integration by substitution work for more complex expressions?

**Tip:** Always check if you can simplify the integrand before integrating, as this often makes the process easier.

Evaluate the given indefinite integrals: 
a) ∫10 sin x dx = 10 cos(x) + C.
b) ∫8 sec^2 s ds = 
c) ∫(15 sec t tan t + 17 csc t cot t) dt = + C.

Explore solutions for evaluating indefinite integrals involving trigonometric functions such as sin, sec, tan, and csc. Learn how to apply basic trigonometric integration formulas in solving problems for high school calculus students.

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