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 }

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