The integral shown in the image is:

$\int \sin(5x) \, dx$

Solution:

To solve this integral, we use substitution:

1. Let $u = 5x$, so $du = 5 \, dx$ or $dx = \frac{du}{5}$.

2. Substitute $u$ and $dx$ into the integral: $\int \sin(5x) \, dx = \int \sin(u) \cdot \frac{1}{5} \, du$

3. Take the constant $\frac{1}{5}$ outside of the integral: $\frac{1}{5} \int \sin(u) \, du$

4. The integral of $\sin(u)$ is $-\cos(u)$: $\frac{1}{5} \cdot (-\cos(u)) + C = -\frac{1}{5} \cos(u) + C$

5. Substitute back $u = 5x$ to get the final answer: $-\frac{1}{5} \cos(5x) + C$

Final Answer:

$\int \sin(5x) \, dx = -\frac{1}{5} \cos(5x) + C$

Do you want further details or examples? Let me know!

Here are 5 related questions to explore:

1. What is the integral of $\cos(5x) \, dx$?
2. How would the solution change if the integrand was $\sin(ax)$?
3. What is the derivative of the result $-\frac{1}{5} \cos(5x) + C$ to verify the answer?
4. How does substitution help simplify integration problems like this?
5. How can definite integrals be applied to functions like $\sin(5x)$?

Tip: When integrating trigonometric functions with coefficients inside the argument, always consider substitution to simplify your calculations!