Key Concepts and Formulas

• Standard Integral Form: The integral of the form $\int \frac{dx}{\sqrt{a^2 - x^2}}$ is $\sin^{-1}\left(\frac{x}{a}\right) + C$.
• Definite Integral Evaluation: The definite integral $\int_b^a f(x)dx$ is evaluated as $F(a) - F(b)$, where $F(x)$ is the antiderivative of $f(x)$.
• Properties of Square Roots: $\sqrt{k \cdot m} = \sqrt{k} \cdot \sqrt{m}$ for non-negative $k, m$.
• Values of Inverse Trigonometric Functions: Knowledge of standard values like $\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3}$ and $\sin^{-1}\left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4}$.

Step-by-Step Solution

We need to evaluate the definite integral: $I = \int\limits_{{{3\sqrt 2 } \over 4}}^{{{3\sqrt 3 } \over 4}} {{{48} \over {\sqrt {9 - 4{x^2}} }}dx}$

Step 1: Manipulate the Integrand to Match the Standard Form

The integrand contains $\sqrt{9 - 4x^2}$. To use the standard integral formula $\int \frac{dx}{\sqrt{a^2 - x^2}}$, the term inside the square root must be in the form $a^2 - x^2$. This means the coefficient of $x^2$ must be $1$.

• Factor out the coefficient of $x^2$: We factor out $4$ from $9 - 4x^2$: $\sqrt{9 - 4x^2} = \sqrt{4\left(\frac{9}{4} - x^2\right)}$
• Apply the square root property: $\sqrt{4\left(\frac{9}{4} - x^2\right)} = \sqrt{4} \cdot \sqrt{\frac{9}{4} - x^2} = 2 \sqrt{\left(\frac{3}{2}\right)^2 - x^2}$
• Rewrite the integrand: Substitute this back into the original integrand: $\frac{48}{\sqrt{9 - 4x^2}} = \frac{48}{2 \sqrt{\left(\frac{3}{2}\right)^2 - x^2}} = \frac{24}{\sqrt{\left(\frac{3}{2}\right)^2 - x^2}}$ The integral now becomes: $I = \int\limits_{{{3\sqrt 2 } \over 4}}^{{{3\sqrt 3 } \over 4}} {{{24} \over {\sqrt {\left(\frac{3}{2}\right)^2 - x^2} }}dx}$

Step 2: Apply the Integration Formula

The integral is now in the form $24 \int \frac{dx}{\sqrt{a^2 - x^2}}$, where $a = \frac{3}{2}$. Using the standard formula $\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\left(\frac{x}{a}\right) + C$: $I = 24 \left[ \sin^{-1}\left(\frac{x}{\frac{3}{2}}\right) \right]_{{{3\sqrt 2 } \over 4}}^{{{3\sqrt 3 } \over 4}}$ $I = 24 \left[ \sin^{-1}\left(\frac{2x}{3}\right) \right]_{{{3\sqrt 2 } \over 4}}^{{{3\sqrt 3 } \over 4}}$

Step 3: Evaluate the Definite Integral Using the Limits

We apply the Fundamental Theorem of Calculus, $F(b) - F(a)$: $I = 24 \left[ \sin^{-1}\left(\frac{2 \cdot \frac{3\sqrt{3}}{4}}{3}\right) - \sin^{-1}\left(\frac{2 \cdot \frac{3\sqrt{2}}{4}}{3}\right) \right]$ Simplify the arguments of the $\sin^{-1}$ function:

• For the upper limit: $\frac{2 \cdot \frac{3\sqrt{3}}{4}}{3} = \frac{\frac{6\sqrt{3}}{4}}{3} = \frac{3\sqrt{3}}{2 \cdot 3} = \frac{\sqrt{3}}{2}$
• For the lower limit: $\frac{2 \cdot \frac{3\sqrt{2}}{4}}{3} = \frac{\frac{6\sqrt{2}}{4}}{3} = \frac{3\sqrt{2}}{2 \cdot 3} = \frac{\sqrt{2}}{2}$ Substitute these simplified values back into the expression for $I$: $I = 24 \left[ \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) - \sin^{-1}\left(\frac{\sqrt{2}}{2}\right) \right]$

Step 4: Calculate the Values of Inverse Trigonometric Functions

We use the known values of the inverse sine function:

• $\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3}$
• $\sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4}$

Substitute these values: $I = 24 \left[ \frac{\pi}{3} - \frac{\pi}{4} \right]$

Step 5: Final Calculation

Perform the subtraction of fractions and the final multiplication: $I = 24 \left[ \frac{4\pi - 3\pi}{12} \right]$ $I = 24 \left[ \frac{\pi}{12} \right]$ $I = \frac{24\pi}{12}$ $I = 2\pi$

Common Mistakes & Tips

• Coefficient of $x^2$: Always ensure the coefficient of $x^2$ inside the square root is $1$ before applying the standard integral formula. Failure to do so will lead to an incorrect value of $a$.
• Constant Factor: Don't forget to carry the constant factor (48 and then 24) through the entire integration and evaluation process.
• Fraction Arithmetic: Be meticulous with fraction arithmetic, especially when subtracting angles in radians.

Summary

The integral was evaluated by first manipulating the integrand to match the standard form of $\int \frac{dx}{\sqrt{a^2 - x^2}}$. After identifying $a = \frac{3}{2}$ and factoring out the constant $24$, the integral was found to be $24 \sin^{-1}\left(\frac{2x}{3}\right)$. Applying the limits of integration and using the known values of inverse trigonometric functions $\sin^{-1}(\frac{\sqrt{3}}{2}) = \frac{\pi}{3}$ and $\sin^{-1}(\frac{\sqrt{2}}{2}) = \frac{\pi}{4}$, the definite integral was calculated to be $2\pi$.

The final answer is $\boxed{2\pi}$, which corresponds to option (A).