The integral shown in the image is:

$\int 2(2x + 1)^2 \, dx$

Solution:

1. Step 1: Expand the expression inside the integral.

We need to expand $(2x + 1)^2$: $(2x + 1)^2 = (2x + 1)(2x + 1) = 4x^2 + 4x + 1$

Now, substitute this back into the integral:

$\int 2(4x^2 + 4x + 1) \, dx$

2. Step 2: Distribute the constant 2.

$\int (8x^2 + 8x + 2) \, dx$

3. Step 3: Integrate term by term.

Now, integrate each term separately:

$\int 8x^2 \, dx = \frac{8x^3}{3}$ $\int 8x \, dx = 4x^2$ $\int 2 \, dx = 2x$

4. Step 4: Combine the results.

$\frac{8x^3}{3} + 4x^2 + 2x + C$

Final Answer:

$\frac{8x^3}{3} + 4x^2 + 2x + C$

Would you like further details or clarification on any step?

Relative Questions:

1. What is the method for expanding binomials like $(2x + 1)^2$?
2. How do you perform integration on polynomial functions?
3. What is the general rule for integrating $x^n$?
4. How would you handle an integral involving a square root instead of a square?
5. How do you apply substitution in integrals like this?

Tip: Always simplify expressions before integrating to make the process easier.