3.6 Q-6

Question Statement

Evaluate the integral: $\int_{2}^{\sqrt{5}} x \sqrt{x^2 - 1} , dx$

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Background and Explanation

This is a standard integral involving a square root of a quadratic expression. In problems like this, we often use substitution and the power rule for integration to simplify the expression.

To solve this, we will apply a standard technique for handling integrals of the form $x \sqrt{x^2 - 1}$, which often involves recognizing patterns and using substitution methods.

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Solution

1. Rewriting the Integral:

   The given integral is: $\int_{2}^{\sqrt{5}} x \sqrt{x^2 - 1} , dx$

   We can simplify the expression by recognizing that $x \sqrt{x^2 - 1}$ can be handled using a substitution method, making the integral easier to solve.

2. Substitute for Simplicity:

   First, apply the substitution $u = x^2 - 1$, so that $du = 2x , dx$. This will help simplify the integral.

   The limits of integration change accordingly:

   • When $x = 2$, $u = 2^2 - 1 = 3$
   • When $x = \sqrt{5}$, $u = (\sqrt{5})^2 - 1 = 4$

   Therefore, the integral becomes: $\int_{3}^{4} \frac{1}{2} \sqrt{u} , du$

3. Integrate Using the Power Rule:

   The integral $\int \sqrt{u} , du$ is a standard integral, and we can apply the power rule for integration: $\int u^n , du = \frac{u^{n+1}}{n+1}$

   For $\sqrt{u} = u^{1/2}$, we have: $\int u^{1/2} , du = \frac{u^{3/2}}{3/2} = \frac{2}{3} u^{3/2}$

4. Evaluate the Integral:

   Now we apply the limits of integration: $\frac{1}{2} \times \frac{2}{3} \left[ u^{3/2} \right]_{3}^{4}$

   Substituting the limits: $= \frac{1}{3} \left[ (4)^{3/2} - (3)^{3/2} \right]$

   Simplify the terms:

   • $(4)^{3/2} = 2^3 = 8$
   • $(3)^{3/2} = 3\sqrt{3}$

   Therefore, we have: $= \frac{1}{3} \left[ 8 - 3\sqrt{3} \right]$

   The final result is: $\boxed{\frac{1}{3} \left[ 8 - 3\sqrt{3} \right]}$

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Key Formulas or Methods Used

• Substitution Method:
  If $u = x^2 - 1$, then $du = 2x , dx$.

• Power Rule for Integration:
  $\int u^n , du = \frac{u^{n+1}}{n+1}$

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Summary of Steps

1. Recognize the integral $x \sqrt{x^2 - 1}$ and apply substitution: $u = x^2 - 1$.
2. Change the limits of integration to match the substitution.
3. Use the power rule to integrate:
   $\int \sqrt{u} , du = \frac{2}{3} u^{3/2}$
4. Apply the limits of integration and simplify the result.
5. The final answer is: $\boxed{\frac{1}{3} \left[ 8 - 3\sqrt{3} \right]}$