The problem in the image involves evaluating the integral:

$I = \int 2x \cos(x^2 - 5) \, dx$

Solution:

1. Substitution: Let $u = x^2 - 5$.
   Then,
   $du = 2x \, dx$ This simplifies $2x \, dx$ to $du$.

2. Rewriting the Integral: Substituting $u$ and $du$ into the integral: $I = \int \cos(u) \, du$

3. Integration: The integral of $\cos(u)$ is: $\int \cos(u) \, du = \sin(u) + C$ where $C$ is the constant of integration.

4. Substitute Back: Replace $u$ with $x^2 - 5$: $I = \sin(x^2 - 5) + C$

Final Answer:

$\int 2x \cos(x^2 - 5) \, dx = \sin(x^2 - 5) + C$

Let me know if you need further explanation!

Related Questions:

1. What is the general formula for solving integrals of the form $\int f(ax^2 + b) \, x dx$?
2. How does substitution simplify complex integrals?
3. Can we solve this integral using integration by parts instead? Why or why not?
4. What is the integral of $\sin(x^2)$ or similar functions?
5. How does the chain rule relate to solving integrals using substitution?

Tip:

Always double-check your substitution steps and limits (if it's a definite integral) to ensure accuracy!