3.6 Q-9

Question Statement

Evaluate the integral: $\int_{-1}^{1} \left( x + \frac{1}{2} \right) \sqrt{x^2 + x + 1} , dx$

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Background and Explanation

This problem requires the integration of a product of a linear function and a square root function. The technique used here is based on simplifying the integral by manipulating the expression and applying known integral formulas. Familiarity with the basic rules of integration, particularly for powers and square roots, is essential. You should also recognize that this integral involves symmetry, as the limits are from -1 to 1.

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Solution

1. Rewrite the Integral:

   Start by rewriting the integral for simplicity: $\int_{-1}^{1} \left( x + \frac{1}{2} \right) \sqrt{x^2 + x + 1} , dx$

   Now, distribute the terms inside the parentheses: $= \int_{-1}^{1} \left[ \left( x + \frac{1}{2} \right) \sqrt{x^2 + x + 1} \right] , dx$

   Factor out the constant $\frac{1}{2}$: $= \frac{1}{2} \int_{-1}^{1} (2x + 1) \sqrt{x^2 + x + 1} , dx$

2. Apply a Simple Substitution:

   Here, we apply a direct method of simplifying the integrand. Observe that the square root term suggests a straightforward substitution. $= \frac{1}{2} \int_{-1}^{1} \left( x^2 + x + 1 \right)^{\frac{1}{2}} (2x + 1) , dx$

3. Integral Simplification Using a Known Formula:

   The integral now has the form suitable for the following formula: $\int \left( x^2 + x + 1 \right)^{\frac{1}{2}} , dx$

   We can solve this integral using a known formula for powers of quadratic expressions. $= \frac{1}{2} \times \frac{2}{3} \left( x^2 + x + 1 \right)^{\frac{3}{2}} \Bigg|_{-1}^{1}$

4. Evaluate the Integral at the Limits:

   Substitute the upper and lower limits (1 and -1) into the expression: $= \frac{1}{3} \left[ \left( (1)^2 + (1) + 1 \right)^{\frac{3}{2}} - \left( (-1)^2 - (1) + 1 \right)^{\frac{3}{2}} \right]$

   Simplifying inside the brackets:

   • For $x = 1$:
     $(1^2 + 1 + 1) = 3 \quad \Rightarrow \quad 3^{\frac{3}{2}} = \sqrt{3}^3 = 3\sqrt{3}$

   • For $x = -1$:
     $((-1)^2 - 1 + 1) = 1 \quad \Rightarrow \quad 1^{\frac{3}{2}} = 1$

5. Final Result:

   Substitute these values back into the equation: $= \frac{1}{3} \left[ 3\sqrt{3} - 1 \right]$

   So the final result is: $= \boxed{\sqrt{3} - \frac{1}{3}}$

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Key Formulas or Methods Used

• Integration of Square Root Functions:
  The integral formula used here is: $\int \left( x^2 + x + 1 \right)^{\frac{1}{2}} , dx$

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Summary of Steps

1. Rewrite the integral and simplify:
   $\int_{-1}^{1} \left( x + \frac{1}{2} \right) \sqrt{x^2 + x + 1} , dx$
   becomes
   $\frac{1}{2} \int_{-1}^{1} (2x + 1) \sqrt{x^2 + x + 1} , dx$

2. Apply substitution and simplify the integral.

3. Use the known formula for the square root of a quadratic expression.

4. Substitute the limits of integration and simplify to get the final result.

5. The result is:
   $\boxed{\sqrt{3} - \frac{1}{3}}$