Evaluate Integral from 0 to 11 StartFraction x cubed dx Over x squared plus 22 x plus 121 EndFraction
.
Question content area bottom
Part 1
Integral from 0 to 11 StartFraction x cubed dx Over x squared plus 22 x plus 121 EndFraction
equals
  
enter your response here
​(Simplify your answer. Type an exact​ answer.)

We are tasked with evaluating the integral:

\[
I = \int_0^{11} \frac{x^3}{x^2 + 22x + 121} \, dx
\]

### Step 1: Simplify the denominator

Notice that the denominator can be rewritten as a perfect square:
\[
x^2 + 22x + 121 = (x + 11)^2
\]
Thus, the integral becomes:

\[
I = \int_0^{11} \frac{x^3}{(x + 11)^2} \, dx
\]

### Step 2: Split \(x^3\) using polynomial division

We divide \(x^3\) by \((x + 11)^2\). Using polynomial long division, we divide \(x^3\) by \((x + 11)\), which gives:

\[
x^3 \div (x + 11) = x^2 - 11x + 121 \quad \text{(remainder } 0)
\]
So, we rewrite the integrand as:

\[
\frac{x^3}{(x + 11)^2} = x^2 - 11x + 121
\]

### Step 3: Separate the integral

Now, we can separate the integral into three parts:

\[
I = \int_0^{11} \left( x^2 - 11x + 121 \right) \, dx
\]

### Step 4: Evaluate each term

We now evaluate each term individually:

1. \(\int_0^{11} x^2 \, dx\):

\[
\int x^2 \, dx = \frac{x^3}{3}, \quad \text{so} \quad \left[ \frac{x^3}{3} \right]_0^{11} = \frac{11^3}{3} = \frac{1331}{3}
\]

2. \(\int_0^{11} -11x \, dx\):

\[
\int -11x \, dx = -\frac{11x^2}{2}, \quad \text{so} \quad \left[ -\frac{11x^2}{2} \right]_0^{11} = -\frac{11(11^2)}{2} = -\frac{11 \times 121}{2} = -\frac{1331}{2}
\]

3. \(\int_0^{11} 121 \, dx\):

\[
\int 121 \, dx = 121x, \quad \text{so} \quad \left[ 121x \right]_0^{11} = 121 \times 11 = 1331
\]

### Step 5: Combine the results

Now, sum the three parts:

\[
I = \frac{1331}{3} - \frac{1331}{2} + 1331
\]

To simplify, we need a common denominator:

\[
I = \frac{1331}{3} - \frac{1331 \times 2}{4} + \frac{1331 \times 3

Evaluate the integral from 0 to 11 of (x^3 / (x^2 + 22x + 121)) dx. Simplify your answer and provide an exact result.

Learn how to evaluate the integral of x^3 / (x^2 + 22x + 121) from 0 to 11 using polynomial division and calculus techniques. This detailed solution simplifies the expression and walks through each step of the integration process. Suitable for advanced high school or college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation