https://file.aiquickdraw.com/mathbot/file/1726698643816mv3i8n7x.jpg

The integral you've provided is:

\[
\int_0^1 \frac{x^2 + x + 1}{(x+1)^2 (x+2)} \, dx
\]

To solve this integral, we'll start by attempting partial fraction decomposition to simplify the integrand. The expression decomposes as follows:

\[
\frac{x^2 + x + 1}{(x+1)^2 (x+2)} = \frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{x+2}
\]

Let's find constants \(A\), \(B\), and \(C\).

1. **Setting up the equation**:
\[
x^2 + x + 1 = A(x+1)(x+2) + B(x+2) + C(x+1)^2
\]
   
2. **Expanding and simplifying**:
\[
x^2 + x + 1 = (A+C)x^2 + (3A+2B+3C)x + (2A+2B+C)
\]

3. **Equating coefficients**:
   - For \(x^2\): \(A + C = 1\)
   - For \(x\): \(3A + 2B + 3C = 1\)
   - For the constant term: \(2A + 2B + C = 1\)

Now, let's solve for \(A\), \(B\), and \(C\).The constants for the partial fraction decomposition are found to be \(A = 2\), \(B = -1\), and \(C = -1\). Substituting these values back into the decomposition gives us:

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

Now, let's integrate each term separately over the interval from 0 to 1:

\[
\int_0^1 \left( \frac{2}{x+1} - \frac{1}{(x+1)^2} - \frac{1}{x+2} \right) \, dx
\]

I'll calculate this integral now.The value of the integral is approximately \(0.481\).

If you need further details or have any questions about the steps, please let me know!

Here are some additional questions you might consider:
1. How would you solve the integral if the limits were changed?
2. What is the importance of using partial fractions in solving integrals?
3. How does the presence of multiple linear factors in the denominator affect the partial fraction decomposition?
4. What other methods could be used to integrate functions similar to this one?
5. How can integral calculus be applied to real-world problems?

**Tip:** Always verify the domain of the function before integrating, especially when dealing with rational functions, to avoid integrating over singularities.

Solve the integral: ∫[0 to 1] (x^2 + x + 1) / ((x + 1)^2 (x + 2)) dx

Solve the integral of (x^2 + x + 1) / ((x+1)^2 (x+2)) from 0 to 1 using partial fraction decomposition. This step-by-step solution includes finding constants for the decomposition and integrating each term individually.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation