https://file.aiquickdraw.com/mathbot/file/1738500101021ic1kpyc1.jpg

The integral shown in the image is:

\[
\int \frac{1}{2} x (2^{x^2}) \, dx
\]

To solve this, we can use substitution and a method known as "integration by parts," depending on your familiarity with these techniques.

### Step 1: Substitution
Let \( u = x^2 \). This gives:

\[
du = 2x \, dx
\]

Then, we can rewrite the integral as:

\[
\frac{1}{2} \int 2x \cdot 2^{x^2} \, dx
\]

This simplifies to:

\[
\frac{1}{2} \int 2^{u} \, du
\]

### Step 2: Solving the integral
Now, we can integrate \( 2^u \). The general form of the integral of \( a^u \) is:

\[
\int a^u \, du = \frac{a^u}{\ln(a)}
\]

So:

\[
\int 2^u \, du = \frac{2^u}{\ln(2)}
\]

### Step 3: Substituting back
Substitute back \( u = x^2 \):

\[
\frac{1}{2} \cdot \frac{2^{x^2}}{\ln(2)} + C
\]

### Final Answer:
\[
\frac{2^{x^2}}{2 \ln(2)} + C
\]

Where \( C \) is the constant of integration.

Would you like a further breakdown of any of the steps? Or would you like me to explore a related topic? 

Here are a few additional questions for you:

1. How do you perform a substitution when solving integrals?
2. What is the role of logarithmic functions in solving exponential integrals?
3. Can you solve the same integral using integration by parts?
4. How do you evaluate definite integrals once the antiderivative is found?
5. What is the relationship between \( e^x \) and \( 2^x \) in integration?

**Tip:** When solving integrals involving exponential functions, substitution is a powerful tool to simplify the expression before integrating.

Learn how to solve the integral ∫ (1/2) x (2^(x^2)) dx using substitution and integration by parts. This step-by-step solution covers key concepts of exponential functions, substitution, and integration techniques. Suitable for students in Grades 11-12.

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