To find the antiderivative of the function \( 2 \sin(2x^2) + 2 \), we approach it term by term:

### 1. Antiderivative of \( 2 \sin(2x^2) \):
To integrate this term, we can use **substitution**.

Let:
\[
u = 2x^2 \quad \Rightarrow \quad \frac{du}{dx} = 4x \quad \Rightarrow \quad dx = \frac{du}{4x}
\]

Thus, the term becomes:
\[
2 \sin(2x^2) dx = \frac{1}{2x} \cdot \int \sin(u) du
\]

However, this approach doesn’t seem to lead to an easily integrable form using elementary functions. Therefore, this integral likely requires advanced techniques or cannot be expressed in terms of elementary functions.

### 2. Antiderivative of \( 2 \):
\[
\int 2 \, dx = 2x
\]

### Final Answer:
The antiderivative of \( 2 \sin(2x^2) + 2 \) involves an integral for which no elementary antiderivative exists. Therefore, the final answer is:
\[
\int \left( 2 \sin(2x^2) + 2 \right) dx = \int 2 \sin(2x^2) dx + 2x + C
\]
The first integral may be left in its original form or solved numerically.

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Would you like further details on any part of this process, or do you have any other questions?

### 5 Related Questions:
1. How do we apply the substitution method in more complex integrals?
2. What are elementary functions, and why can't some integrals be expressed using them?
3. How can we numerically integrate functions like \( 2 \sin(2x^2) \)?
4. Are there any special functions that can be used to express non-elementary integrals?
5. What is the difference between definite and indefinite integrals?

### Tip:
For integrals that cannot be expressed in terms of elementary functions, advanced methods such as numerical integration or using special functions like the Fresnel or error functions might be necessary.

Explore the process of finding the antiderivative of 2sin(2x^2) + 2 using the substitution method. This problem involves advanced calculus concepts like trigonometric functions and integrals that cannot be expressed with elementary functions. Suitable for Grades 11-12 or college-level students.

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