https://file.aiquickdraw.com/mathbot/file/1726762648020pinob6c3.jpg

The integral given in the image is:

\[
\int_0^2 7x^3 \sqrt{x^2 + 4} \, dx
\]

Let's solve this step by step.

### Step 1: Substitution
Let’s use the substitution method to simplify the square root term. Set:
\[
u = x^2 + 4
\]
Then, the differential of \( u \) with respect to \( x \) is:
\[
du = 2x \, dx \quad \Rightarrow \quad \frac{du}{2} = x \, dx
\]

We need to change the limits of integration as well:
- When \( x = 0 \), \( u = 0^2 + 4 = 4 \),
- When \( x = 2 \), \( u = 2^2 + 4 = 8 \).

Now we can rewrite the integral. First, express \( x^3 \) in terms of \( u \):
\[
x^2 = u - 4 \quad \Rightarrow \quad x^3 = x(u-4)^{3/2} \quad
}


The

Evaluate the integral: ∫ from 0 to 2 of 7x^3 √(x^2 + 4) dx

Learn how to evaluate the integral ∫ from 0 to 2 of 7x^3 √(x^2 + 4) dx using the substitution method. This solution explains key calculus concepts, including substitution and changing limits of integration. Ideal for college-level calculus students.

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