https://r2.erweima.ai/mathbot/image/1725938163727du9yfzx9.jpg

The integral provided in the image is:

\[
\int (3\cos{x} + 5\sin{x})\, dx
\]

We can solve this integral term by term:

1. The integral of \( \cos{x} \) is \( \sin{x} \), so:
   \[
   \int 3\cos{x} \, dx = 3\sin{x}
   \]

2. The integral of \( \sin{x} \) is \( -\cos{x} \), so:
   \[
   \int 5\sin{x} \, dx = -5\cos{x}
   \]

Thus, the complete integral is:

\[
\int (3\cos{x} + 5\sin{x}) \, dx = 3\sin{x} - 5\cos{x} + C
\]

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

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How do you apply integration techniques to more complex trigonometric functions?
2. What happens when you differentiate the result of this integral?
3. How can you integrate products of trigonometric functions, such as \( \cos(x)\sin(x) \)?
4. What are the common integrals of trigonometric functions, and how can they be applied?
5. How do you evaluate definite integrals involving trigonometric functions?

**Tip:** When integrating trigonometric functions, knowing the basic integral identities helps speed up the process.

Learn how to find the integral of the expression (3cos(x) + 5sin(x))dx, covering integration techniques of basic trigonometric functions. This detailed explanation includes the solution and related integral formulas, suitable for Grades 10-12.

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