3.6 Q-12

Question Statement

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

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Background and Explanation

This is an integral involving a rational expression with a square root and a term involving $\frac{1}{x^2}$. To simplify this integral, we’ll use substitution to convert it into a more manageable form. Specifically, we will let: $y = x + \frac{1}{x}$

This substitution will simplify the integrand and allow us to proceed with the integration.

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Solution

1. Substitute the Expression:

   Let: $y = x + \frac{1}{x}$

   Now, differentiate with respect to $x$ to find $dy$: $dy = \left( 1 - \frac{1}{x^2} \right) dx$

2. Adjust Limits of Integration:

   When $x = 1$, we get: $y = 1 + \frac{1}{1} = 2$

   When $x = 2$, we get: $y = 2 + \frac{1}{2} = \frac{5}{2}$

   Thus, the limits for $y$ change from $x = 1$ to $x = 2$, corresponding to $y = 2$ to $y = \frac{5}{2}$.

3. Rewrite the Integral:

   Substitute into the original integral: $\int_{1}^{2} \left(x + \frac{1}{x}\right)^{\frac{1}{2}} \left(1 - \frac{1}{x^2}\right) dx$ becomes: $\int_{2}^{\frac{5}{2}} y^{\frac{1}{2}} , dy$

4. Integrate:

   Now, integrate the function $y^{\frac{1}{2}}$ with respect to $y$. The formula for the integral of $y^n$ is: $\int y^n , dy = \frac{y^{n+1}}{n+1}$

   In this case, $n = \frac{1}{2}$, so: $\int y^{\frac{1}{2}} , dy = \frac{2}{3} y^{\frac{3}{2}}$

5. Evaluate the Integral:

   Now, substitute the limits $\frac{5}{2}$ and $2$ into the antiderivative: $\frac{2}{3} \left[ y^{\frac{3}{2}} \right]_{2}^{\frac{5}{2}}$

   This becomes: $\frac{2}{3} \left[ \left( \frac{5}{2} \right)^{\frac{3}{2}} - 2^{\frac{3}{2}} \right]$

6. Simplify the Result:

   Now simplify the expression: $\frac{2}{3} \left[ \frac{5^{\frac{3}{2}}}{2^{\frac{3}{2}}} - 2^{\frac{3}{2}} \right]$

   Factor out $\frac{2}{3 \sqrt{2}}$ to make the expression easier to handle: $\frac{2}{3 \sqrt{2}} \left[ 5 \sqrt{5} - 8 \right]$

   Thus, the value of the integral is: $\boxed{\frac{2}{3 \sqrt{2}} \left[ 5 \sqrt{5} - 8 \right]}$

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Key Formulas or Methods Used

• Substitution: $y = x + \frac{1}{x}$ $dy = \left( 1 - \frac{1}{x^2} \right) dx$

• Integral of a Power: $\int y^n , dy = \frac{y^{n+1}}{n+1}$

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Summary of Steps

1. Let $y = x + \frac{1}{x}$, and differentiate to get $dy = \left(1 - \frac{1}{x^2}\right) dx$.

2. Change the limits of integration: when $x = 1$, $y = 2$; and when $x = 2$, $y = \frac{5}{2}$.

3. Substitute into the integral and simplify it to $\int_{2}^{\frac{5}{2}} y^{\frac{1}{2}} , dy$.

4. Integrate using the formula $\int y^n , dy = \frac{y^{n+1}}{n+1}$ to get $\frac{2}{3} y^{\frac{3}{2}}$.

5. Evaluate the result at the limits to get: $\frac{2}{3 \sqrt{2}} \left[ 5 \sqrt{5} - 8 \right]$

6. Final answer: $\boxed{\frac{2}{3 \sqrt{2}} \left[ 5 \sqrt{5} - 8 \right]}$