3.6 Q-5

Question Statement

Evaluate the integral: $\int_{1}^{\sqrt{5}} \sqrt{(2t - 1)^{3}} , dx$

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Background and Explanation

In this problem, we are tasked with integrating a function involving a square root of a cubic expression. The integrand is of the form $\sqrt{(2t - 1)^3}$, which requires manipulation using substitution and integration techniques. We’ll use the power rule to simplify and solve this integral.

The square root function can be written as a power expression $(2t - 1)^{3/2}$, and we need to adjust the expression accordingly for integration.

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Solution

1. Rewriting the Integral:

   The given integral is: $\int_{1}^{\sqrt{5}} \sqrt{(2t - 1)^{3}} , dx$

   First, simplify the square root: $\sqrt{(2t - 1)^{3}} = (2t - 1)^{3/2}$

   Therefore, the integral becomes: $\int_{1}^{\sqrt{5}} (2t - 1)^{3/2} , dx$

2. Introduce a Constant Factor:

   We can factor out constants to simplify the integral. Notice that $2$ is a constant inside the integrand: $\frac{1}{2} \int_{1}^{\sqrt{5}} (2t - 1)^{3/2} \cdot 2 , dx$

   Now the integral becomes: $\frac{1}{2} \cdot 2 \int_{1}^{\sqrt{5}} (2t - 1)^{3/2} , dx$

   The $2$‘s cancel out, leaving: $\int_{1}^{\sqrt{5}} (2t - 1)^{3/2} , dx$

3. Apply the Power Rule for Integration:

   We can now apply the power rule for integration. The general rule is: $\int (u^n) , du = \frac{u^{n+1}}{n+1}$

   For $(2t - 1)^{3/2}$, we have $n = 3/2$, so the integral becomes: $\frac{(2t - 1)^{5/2}}{5/2}$

   Therefore, we can now write: $\frac{1}{2} \cdot \frac{2}{5} \left[ (2t - 1)^{5/2} \right]_{1}^{\sqrt{5}}$

4. Evaluate the Integral at the Limits:

   Now we evaluate the expression at the limits $t = 1$ and $t = \sqrt{5}$: $\frac{1}{5} \left[ (2\sqrt{5} - 1)^{5/2} - (2(1) - 1)^{5/2} \right]$

   Simplify each term:

   • $2\sqrt{5} - 1 = 2\sqrt{5} - 1$
   • $2(1) - 1 = 1$

   So we get: $\frac{1}{5} \left[ (2\sqrt{5} - 1)^{5/2} - 1^{5/2} \right]$

5. Final Expression:

   Therefore, the final result is: $\frac{1}{5} \left[ (2\sqrt{5} - 1)^{5/2} - 1 \right]$

Thus, the value of the integral is: $\boxed{\frac{1}{5} \left[ (2\sqrt{5} - 1)^{5/2} - 1 \right]}$

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Key Formulas or Methods Used

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

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Summary of Steps

1. Rewrite the integrand as $(2t - 1)^{3/2}$.
2. Factor out constants and simplify the integral.
3. Apply the power rule for integration: $\int (2t - 1)^{3/2} , dx = \frac{2}{5} (2t - 1)^{5/2}$
4. Evaluate the integral at the limits $t = 1$ and $t = \sqrt{5}$: $\frac{1}{5} \left[ (2\sqrt{5} - 1)^{5/2} - 1 \right]$
5. The final result is: $\boxed{\frac{1}{5} \left[ (2\sqrt{5} - 1)^{5/2} - 1 \right]}$