To evaluate the indefinite integral $\int 2x \cos(2x^2) \, dx$ using substitution, follow these steps:

Step 1: Choose the substitution

Let's choose the substitution: $u = 2x^2$ Then, differentiate $u$ with respect to $x$: $\frac{du}{dx} = 4x$ Thus, $du = 4x \, dx$ or equivalently, $\frac{1}{2} du = 2x \, dx$

Step 2: Substitute in the integral

Substituting $u$ and $du$ into the integral: $\int 2x \cos(2x^2) \, dx = \int \cos(u) \cdot \frac{1}{2} \, du$

Step 3: Integrate with respect to $u$

Now, integrate $\cos(u)$ with respect to $u$: $\frac{1}{2} \int \cos(u) \, du = \frac{1}{2} \sin(u) + C$ where $C$ is the constant of integration.

Step 4: Substitute back $u = 2x^2$

Finally, substitute $u = 2x^2$ back into the equation: $\frac{1}{2} \sin(u) + C = \frac{1}{2} \sin(2x^2) + C$

Final Answer:

The indefinite integral is: $\int 2x \cos(2x^2) \, dx = \frac{1}{2} \sin(2x^2) + C$

Would you like more details, or do you have any questions?

Here are some related questions to consider:

1. How would you evaluate the integral $\int x e^{x^2} \, dx$ using substitution?
2. Can you use substitution to evaluate $\int \sin(3x^2) \cdot 6x \, dx$?
3. What is the integral of $\int \cos(3x) \, dx$ without substitution?
4. How would you integrate $\int x \cos(x^2 + 1) \, dx$?
5. How do you determine the best substitution for an integral?

Tip: When performing substitution, always check if the derivative of your substitution matches part of the integrand to simplify the process.