https://file.aiquickdraw.com/mathbot/file/1726368980355mcvv9tdr.jpg

The expression you provided is the integral:

\[
\int (2x - 1) (x^2 - x + 1)^3 \, dx
\]

To solve this, we can use substitution since the expression inside the parentheses \((x^2 - x + 1)\) is raised to the power of 3, and its derivative appears in the form of the first factor \((2x - 1)\).

### Step-by-Step Solution

1. **Substitute**: Let \( u = x^2 - x + 1 \). Then, the derivative of \( u \) with respect to \( x \) is:
   \[
   \frac{du}{dx} = 2x - 1
   \]
   This gives us:
   \[
   du = (2x - 1) dx
   \]
   So, the integral becomes:
   \[
   \int u^3 \, du
   \]

2. **Integrate**: Now, integrate \( u^3 \):
   \[
   \int u^3 \, du = \frac{u^4}{4} + C
   \]

3. **Substitute Back**: Replace \( u \) with \( x^2 - x + 1 \):
   \[
   \frac{(x^2 - x + 1)^4}{4} + C
   \]

Thus, the solution to the integral is:

\[
\frac{(x^2 - x + 1)^4}{4} + C
\]

Would you like further details on any step, or do you have any questions?

---

Here are five questions related to this topic:

1. How do we choose the substitution for integrals involving polynomials?
2. Why is the chain rule important in solving integrals by substitution?
3. Can this method be used for other powers of polynomials in integrals?
4. How do we verify if an integral solution is correct?
5. What happens if we cannot find an obvious substitution for the integral?

**Tip**: Always look for a factor that resembles the derivative of another part of the integrand when using substitution. This simplifies the process significantly.

Find the integral of (2x - 1)(x^2 - x + 1)^3 dx.

This detailed solution shows how to find the integral of (2x - 1)(x^2 - x + 1)^3 using substitution. This method simplifies the polynomial expression and makes the integration process more straightforward. Suitable for students in Grades 11-12 or early college.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation