https://file.aiquickdraw.com/mathbot/file/1727242972424gd8388ey.jpg

I see that the image contains integrals where you need to select an appropriate substitution \( u \) for each. Let’s go through them one by one:

1. **Integral:**
   \[
   \int 3(3x - 1)^4 \, dx
   \]
   Here, the best substitution would be:
   \[
   u = 3x - 1
   \]
   This simplifies the expression inside the parentheses.

2. **Integral:**
   \[
   \int (2x + 1)(x^2 + x) \, dx
   \]
   For this integral, a reasonable choice is:
   \[
   u = x^2 + x
   \]
   since the derivative of \( x^2 + x \) gives a term related to \( 2x + 1 \), which will simplify the integral.

3. **Integral:**
   \[
   \int 3x^2 \sqrt{x^3 - 2} \, dx
   \]
   Here, it's best to set:
   \[
   u = x^3 - 2
   \]
   because the derivative of \( x^3 - 2 \) is \( 3x^2 \), which matches the given expression.

4. **Integral:**
   \[
   \int \frac{-4x}{1 - 2x^2} \, dx
   \]
   For this rational expression, a good choice is:
   \[
   u = 1 - 2x^2
   \]
   because the derivative of \( 1 - 2x^2 \) is \( -4x \), which matches the numerator.

5. **Integral:**
   \[
   \int \cos^2(x) \sin(x) \, dx
   \]
   In this case, the natural choice would be:
   \[
   u = \cos(x)
   \]
   since the derivative of \( \cos(x) \) is \( -\sin(x) \), and this will simplify the integral involving powers of cosine and sine.

---

Would you like a detailed solution for any of these integrals? 

### Here are 5 relative questions that expand on this:
1. How does substitution simplify integrals that contain composite functions?
2. Can substitution be used for trigonometric integrals other than those involving sine and cosine?
3. How do you determine the best \( u \)-substitution when the function inside the integral is complex?
4. What is the relationship between \( u \)-substitution and the chain rule in differentiation?
5. Are there any cases where \( u \)-substitution is not an ideal method for solving an integral?

**Tip:** Always check that the derivative of your substitution matches part of the integrand to ensure the substitution simplifies the integral effectively.

What would you let u = in the following integrals?

This page explains how to solve integrals using the substitution method and the general power rule. It covers key integrals such as ∫3(3x - 1)^4 dx and provides step-by-step substitutions.

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