3.6 Q-7

Question Statement

Evaluate the integral: $\int_{1}^{2} \frac{x}{x^2 + 2} , dx$

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Background and Explanation

To solve this problem, we will use the technique of substitution to simplify the integral. The integral involves a rational function, and a common approach for such integrals is to rewrite the expression in a more manageable form using algebraic manipulations. In this case, we will first manipulate the integral into a form that allows us to use the natural logarithm function for easier integration.

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Solution

1. Rewriting the Integral:

   Start with the given integral: $\int_{1}^{2} \frac{x}{x^2 + 2} , dx$

   Notice that the numerator is $x$, which is the derivative of $x^2 + 2$. This suggests that a substitution will help simplify the integral.

2. Substitute to Simplify the Integral:

   Let’s rewrite the integral as follows: $\frac{1}{2} \int_{1}^{2} \frac{2x}{x^2 + 2} , dx$

   This allows us to recognize that the numerator is the derivative of the denominator. Now, we can proceed by recognizing the standard form for an integral of this type.

3. Apply the Integral Formula:

   The integral of $\frac{2x}{x^2 + 2}$ is a standard result: $\int \frac{2x}{x^2 + 2} , dx = \ln|x^2 + 2|$

   Applying this, we now have: $\frac{1}{2} \left[ \ln|x^2 + 2| \right]_{1}^{2}$

4. Evaluate the Limits:

   Now, substitute the upper and lower limits into the expression: $\frac{1}{2} \left[ \ln(2^2 + 2) - \ln(1^2 + 2) \right]$

   Simplify the terms inside the logarithms: $= \frac{1}{2} \left[ \ln(4 + 2) - \ln(1 + 2) \right]$ $= \frac{1}{2} \left[ \ln(6) - \ln(3) \right]$

5. Simplify the Logarithmic Expression:

   Use the property of logarithms: $\ln(a) - \ln(b) = \ln \left( \frac{a}{b} \right)$

   So we get: $\frac{1}{2} \ln \left( \frac{6}{3} \right)$

   Simplify the fraction: $= \frac{1}{2} \ln(2)$

   Therefore, the final result is: $\boxed{\frac{1}{2} \ln 2}$

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Key Formulas or Methods Used

• Substitution:
  Rewrite the integral to simplify the expression:
  $\frac{1}{2} \int \frac{2x}{x^2 + 2} , dx$

• Standard Integral:
  $\int \frac{2x}{x^2 + 2} , dx = \ln|x^2 + 2|$

• Logarithmic Properties:
  $\ln(a) - \ln(b) = \ln \left( \frac{a}{b} \right)$

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Summary of Steps

1. Rewrite the integral as $\frac{1}{2} \int \frac{2x}{x^2 + 2} , dx$.
2. Use the standard integral $\int \frac{2x}{x^2 + 2} , dx = \ln|x^2 + 2|$.
3. Substitute the limits into the logarithmic expression.
4. Simplify the logarithms using properties of logarithms.
5. The final answer is: $\boxed{\frac{1}{2} \ln 2}$