3.6 Q-24

Question Statement

Evaluate the following definite integral:

$$\int_1^3 \frac{x^2 - 2}{x + 1} , dx$$

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Background and Explanation

This problem involves an algebraic expression inside an integral. The key steps include simplifying the integrand and breaking it down into easier terms. We will use algebraic manipulation and the properties of logarithms to solve the integral. Key concepts used:

• Division of polynomials: Decompose the expression into simpler parts.
• Integration of basic functions: Use basic integration rules for polynomials and logarithmic functions.

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Solution

Step 1: Simplify the integrand

We start by simplifying the numerator $x^2 - 2$. We can rewrite this as:

$$x^2 - 2 = (x^2 - 1) - 1$$

Thus, the integral becomes:

$$\int_1^3 \frac{x^2 - 1 - 1}{x + 1} , dx$$

We can now split this into two separate integrals:

$$\int_1^3 \frac{x^2 - 1}{x + 1} , dx - \int_1^3 \frac{1}{x + 1} , dx$$

Step 2: Simplify further using polynomial division

The term $\frac{x^2 - 1}{x + 1}$ can be simplified by performing polynomial long division. Dividing $x^2 - 1$ by $x + 1$:

1. Divide the leading term $x^2$ by $x$, which gives $x$.
2. Multiply $x$ by $x + 1$, resulting in $x^2 + x$.
3. Subtract this from $x^2 - 1$:

$$(x^2 - 1) - (x^2 + x) = -x - 1$$

4. Divide $-x$ by $x$, which gives $-1$.
5. Multiply $-1$ by $x + 1$, resulting in $-x - 1$.
6. Subtract again:

$$(-x - 1) - (-x - 1) = 0$$

Thus, the division yields:

$$\frac{x^2 - 1}{x + 1} = x - 1$$

So, the integral now becomes:

$$\int_1^3 (x - 1) , dx - \int_1^3 \frac{1}{x + 1} , dx$$

Step 3: Integrate each term

Now, we integrate each term separately.

• First term:

$$\int_1^3 (x - 1) , dx = \int_1^3 x , dx - \int_1^3 1 , dx$$

The integral of $x$ is:

$$\int x , dx = \frac{x^2}{2}$$

And the integral of 1 is:

$$\int 1 , dx = x$$

Evaluating from $1$ to $3$:

$$\left[\frac{x^2}{2}\right]_1^3 - \left[x\right]_1^3 = \left(\frac{9}{2} - \frac{1}{2}\right) - (3 - 1) = \frac{8}{2} - 2 = 4 - 2 = 2$$

• Second term:

$$\int_1^3 \frac{1}{x + 1} , dx = \ln|x + 1|$$

Evaluating from $1$ to $3$:

$$\left[\ln(x + 1)\right]_1^3 = \ln(3 + 1) - \ln(1 + 1) = \ln 4 - \ln 2 = \ln \left(\frac{4}{2}\right) = \ln 2$$

Step 4: Combine the results

Now, combine the results of the two integrals:

$$2 - \ln 2$$

Thus, the value of the integral is:

$$\boxed{2 - \ln 2}$$

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Key Formulas or Methods Used

• Polynomial Division: Used to simplify the rational expression.
• Basic Power Rule for Integration: $\int x^n , dx = \frac{x^{n+1}}{n+1}$ for $n \neq -1$.
• Logarithmic Integration: $\int \frac{1}{x + 1} , dx = \ln(x + 1)$.

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Summary of Steps

1. Simplify the integrand by breaking the expression $x^2 - 2$ into $(x^2 - 1) - 1$.
2. Perform polynomial division to simplify $\frac{x^2 - 1}{x + 1}$ to $x - 1$.
3. Integrate each term separately:
   • $\int (x - 1) , dx$
   • $\int \frac{1}{x + 1} , dx$
4. Evaluate the integrals from $1$ to $3$.
5. Combine the results to find the final answer: $2 - \ln 2$.

The final answer is:

$$\boxed{2 - \ln 2}$$