Evaluate the indefinite integral using substitution: summa 2x cos(2x^2)dx

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.

Learn how to evaluate the indefinite integral \( \int 2x \cos(2x^2) \, dx \) using substitution. Follow a detailed step-by-step solution with explanations of substitution method and integration of cosine function. Suitable for college-level mathematics.

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