https://file.aiquickdraw.com/mathbot/file/1726976672792h4hphg07.jpg

The integral you uploaded is:

\[
\int 4x^3(2x^2 + 1)^{12} \, dx
\]

### Solution:

We can use the **substitution method** for this integral.

Let:
\[
u = 2x^2 + 1
\]

Then, differentiate both sides with respect to \(x\):
\[
du = 4x \, dx
\]

Now, divide by 4 to solve for \(x^3 dx\):
\[
x^3 \, dx = \frac{1}{4} du
\]

Substitute into the integral:
\[
\int 4x^3(2x^2 + 1)^{12} \, dx = \int (u)^{12} \, du
\]

This is a standard power of \(u\)-integral:
\[
\int u^{12} \, du = \frac{u^{13}}{13} + C
\]

Now substitute back \(u = 2x^2 + 1\):
\[
\frac{(2x^2 + 1)^{13}}{13} + C
\]

Thus, the solution to the integral is:
\[
\frac{(2x^2 + 1)^{13}}{13} + C
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. How does substitution work in integrals?
2. What is the importance of \( du \) in substitution?
3. Can this method be used for other forms of polynomial integrals?
4. What is the general rule for integrating powers of binomials?
5. How do you check the result of an integral?

**Tip**: Always remember to substitute back the original variable after integrating when using substitution!

Learn how to solve the integral ∫ 4x^3 (2x^2 + 1)^12 dx using the substitution method. This problem involves applying the substitution rule and basic integral of powers of u. Suitable for college-level Calculus 1 students.

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