Key Concepts and Formulas

• Power Reduction Formula for Cosine: $\cos^2 \theta = \frac{1 + \cos(2\theta)}{2}$. This is essential for reducing even powers of cosine to simpler forms.
• Integration of Trigonometric Functions: $\int \cos(ax) \mathrm{d}x = \frac{1}{a}\sin(ax) + C$.
• Fundamental Theorem of Calculus: $\int_a^b f(x) \mathrm{d}x = F(b) - F(a)$, where $F(x)$ is an antiderivative of $f(x)$.
• Unit Circle Values: Knowledge of trigonometric values for standard angles like $\frac{\pi}{3}$, $\frac{2\pi}{3}$, and $\frac{4\pi}{3}$ is required.

Step-by-Step Solution

We are asked to evaluate the definite integral $\int\limits_0^{\frac{\pi}{3}} \cos ^4 x \mathrm{~d} x$ and express it in the form $\mathrm{a} \pi+\mathrm{b} \sqrt{3}$, where $\mathrm{a}$ and $\mathrm{b}$ are rational numbers. Then we need to find the value of $9 \mathrm{a}+8 \mathrm{b}$.

Step 1: Reduce the power of the integrand $\cos^4 x$. The integrand is $\cos^4 x$. We use the power reduction formula $\cos^2 \theta = \frac{1 + \cos(2\theta)}{2}$ twice. First, rewrite $\cos^4 x$ as $(\cos^2 x)^2$: $\cos^4 x = (\cos^2 x)^2 = \left(\frac{1 + \cos(2x)}{2}\right)^2$

• Why this step? To make the integral tractable, we need to express $\cos^4 x$ in terms of cosines of multiple angles, which can be integrated directly.

Step 2: Expand and simplify the expression for $\cos^4 x$. Expand the squared term: $\cos^4 x = \frac{1}{4} (1 + \cos(2x))^2 = \frac{1}{4} (1 + 2\cos(2x) + \cos^2(2x))$ Now, apply the power reduction formula again to $\cos^2(2x)$, with $\theta = 2x$: $\cos^2(2x) = \frac{1 + \cos(2 \cdot 2x)}{2} = \frac{1 + \cos(4x)}{2}$ Substitute this back into the expression for $\cos^4 x$: $\cos^4 x = \frac{1}{4} \left(1 + 2\cos(2x) + \frac{1 + \cos(4x)}{2}\right)$ Distribute and combine terms: $\cos^4 x = \frac{1}{4} + \frac{2}{4}\cos(2x) + \frac{1}{8} + \frac{1}{8}\cos(4x)$ $\cos^4 x = \left(\frac{1}{4} + \frac{1}{8}\right) + \frac{1}{2}\cos(2x) + \frac{1}{8}\cos(4x)$ $\cos^4 x = \frac{3}{8} + \frac{1}{2}\cos(2x) + \frac{1}{8}\cos(4x)$

• Why this step? This transforms the integrand into a sum of terms that are easily integrable: a constant and cosine functions of multiples of $x$.

Step 3: Integrate the simplified expression. Now, we evaluate the definite integral: $\int\limits_0^{\frac{\pi}{3}} \left(\frac{3}{8} + \frac{1}{2}\cos(2x) + \frac{1}{8}\cos(4x)\right) \mathrm{~d}x$ Integrate each term: $\int \frac{3}{8} \mathrm{~d}x = \frac{3}{8}x$ $\int \frac{1}{2}\cos(2x) \mathrm{~d}x = \frac{1}{2} \cdot \frac{\sin(2x)}{2} = \frac{1}{4}\sin(2x)$ $\int \frac{1}{8}\cos(4x) \mathrm{~d}x = \frac{1}{8} \cdot \frac{\sin(4x)}{4} = \frac{1}{32}\sin(4x)$ So, the antiderivative is: $\left. \left(\frac{3}{8} x + \frac{1}{4} \sin(2x) + \frac{1}{32} \sin(4x)\right) \right|_0^{\frac{\pi}{3}}$

• Why this step? We apply the basic rules of integration to each term of the transformed integrand.

Step 4: Evaluate the definite integral using the limits of integration. Apply the Fundamental Theorem of Calculus: $F(\frac{\pi}{3}) - F(0)$. Evaluate at the upper limit $x = \frac{\pi}{3}$: $\left(\frac{3}{8} \cdot \frac{\pi}{3} + \frac{1}{4} \sin\left(2 \cdot \frac{\pi}{3}\right) + \frac{1}{32} \sin\left(4 \cdot \frac{\pi}{3}\right)\right)$ $= \frac{\pi}{8} + \frac{1}{4} \sin\left(\frac{2\pi}{3}\right) + \frac{1}{32} \sin\left(\frac{4\pi}{3}\right)$ Evaluate at the lower limit $x = 0$: $\left(\frac{3}{8} \cdot 0 + \frac{1}{4} \sin(0) + \frac{1}{32} \sin(0)\right) = 0 + 0 + 0 = 0$

• Why this step? This is the final evaluation of the integral by substituting the upper and lower bounds.

Step 5: Substitute the trigonometric values and simplify. We need the values of $\sin\left(\frac{2\pi}{3}\right)$ and $\sin\left(\frac{4\pi}{3}\right)$. $\sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2}$ $\sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2}$ Substitute these values: $\frac{\pi}{8} + \frac{1}{4} \left(\frac{\sqrt{3}}{2}\right) + \frac{1}{32} \left(-\frac{\sqrt{3}}{2}\right)$ $= \frac{\pi}{8} + \frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{64}$ Combine the terms with $\sqrt{3}$: $= \frac{\pi}{8} + \left(\frac{1}{8} - \frac{1}{64}\right) \sqrt{3}$ $= \frac{\pi}{8} + \left(\frac{8}{64} - \frac{1}{64}\right) \sqrt{3}$ $= \frac{\pi}{8} + \frac{7\sqrt{3}}{64}$

• Why this step? This step involves correctly evaluating trigonometric functions at specific angles and combining like terms to match the required form.

Step 6: Match the result with the given form and find $a$ and $b$. The integral is equal to $\frac{\pi}{8} + \frac{7\sqrt{3}}{64}$. We are given that the integral is equal to $\mathrm{a} \pi+\mathrm{b} \sqrt{3}$. By comparing the two forms, we have: $a = \frac{1}{8}$ $b = \frac{7}{64}$

• Why this step? To identify the rational coefficients $a$ and $b$ by directly comparing the computed integral with the problem's given format.

Step 7: Calculate $9a + 8b$. Now, we calculate $9a + 8b$: $9a + 8b = 9\left(\frac{1}{8}\right) + 8\left(\frac{7}{64}\right)$ $= \frac{9}{8} + \frac{56}{64}$ Simplify the second term: $\frac{56}{64} = \frac{7 \times 8}{8 \times 8} = \frac{7}{8}$. $= \frac{9}{8} + \frac{7}{8}$ $= \frac{9+7}{8} = \frac{16}{8} = 2$

• Why this step? This is the final calculation to find the required numerical value based on the determined values of $a$ and $b$.

Common Mistakes & Tips

• Incorrect Power Reduction: Ensure the power reduction formula is applied correctly, especially when dealing with $\cos^2(2x)$ where the angle is doubled.
• Trigonometric Values: Double-check the values of trigonometric functions for angles like $\frac{2\pi}{3}$ and $\frac{4\pi}{3}$, paying close attention to the quadrant and sign.
• Algebraic Errors: Be careful with fractions and expanding squared terms to avoid arithmetic mistakes.

Summary

The problem was solved by systematically reducing the power of the integrand $\cos^4 x$ using the power reduction formula for cosine. This transformed the integral into a sum of easily integrable terms. After performing the integration and evaluating the definite integral at the given limits, we obtained the result in the form $\mathrm{a} \pi+\mathrm{b} \sqrt{3}$. By comparing this with the computed integral, the values of $a$ and $b$ were identified. Finally, these values were used to calculate $9a + 8b$, yielding the numerical answer.

The final answer is \boxed{2}.