Question

If \[\int_{0}^{\frac{\pi}{3}} \cos^4 x \, dx = a\pi + b\sqrt{3},\]where $a$ and $b$ are rational numbers, then $9a + 8b$ is equal to:

• A. 2
• B. 1
• C. 3
• D. \(\frac{3}{2}\)

Answer 1

The Correct Option is A

To evaluate the integral \(\int_{0}^{\frac{\pi}{3}} \cos^4 x \, dx\), we apply trigonometric identities and integration methods. The integral is given to equal \(a\pi + b\sqrt{3}\), where \(a\) and \(b\) are rational numbers. Our objective is to determine the value of \(9a + 8b\).

Step 1: Apply the power-reduction identity

The identity for \(\cos^2 x\) is:

\[\cos^2 x = \frac{1 + \cos 2x}{2}\]

Consequently, \(\cos^4 x = (\cos^2 x)^2 = \left(\frac{1 + \cos 2x}{2}\right)^2\).

Step 2: Expand the expression

\[ \cos^4 x = \left(\frac{1 + \cos 2x}{2}\right)^2 = \frac{1 + 2\cos 2x + \cos^2 2x}{4} \]

Using the identity \(\cos^2 2x = \frac{1 + \cos 4x}{2}\), we get:

\[ \cos^4 x = \frac{1 + 2\cos 2x + \frac{1 + \cos 4x}{2}}{4} = \frac{3 + 4\cos 2x + \cos 4x}{8} \]

Step 3: Integrate the expression

Integrate each term from \(0\) to \(\frac{\pi}{3}\):

\[ \int_{0}^{\frac{\pi}{3}} \cos^4 x \, dx = \int_{0}^{\frac{\pi}{3}} \frac{3}{8} \, dx + \int_{0}^{\frac{\pi}{3}} \frac{4\cos 2x}{8} \, dx + \int_{0}^{\frac{\pi}{3}} \frac{\cos 4x}{8} \, dx \]

\[ = \frac{3}{8} \cdot \frac{\pi}{3} + \frac{1}{2} \int_{0}^{\frac{\pi}{3}} \cos 2x \, dx + \frac{1}{8} \int_{0}^{\frac{\pi}{3}} \cos 4x \, dx \]

Step 4: Solve the definite integrals

The general integral is \(\int \cos kx \, dx = \frac{\sin kx}{k} + C\).

Therefore:

\[ \int_{0}^{\frac{\pi}{3}} \cos 2x \, dx = \left[\frac{\sin 2x}{2}\right]_{0}^{\frac{\pi}{3}} = \frac{\sin \frac{2\pi}{3}}{2} - \frac{\sin 0}{2} = \frac{\sqrt{3}}{4} \]

\[ \int_{0}^{\frac{\pi}{3}} \cos 4x \, dx = \left[\frac{\sin 4x}{4}\right]_{0}^{\frac{\pi}{3}} = \frac{\sin \frac{4\pi}{3}}{4} - \frac{\sin 0}{4} = -\frac{\sqrt{3}}{8} \]

Substitute these values back:

\[ \int_{0}^{\frac{\pi}{3}} \cos^4 x \, dx = \frac{\pi}{8} + \frac{1}{2} \cdot \frac{\sqrt{3}}{4} + \frac{1}{8} \left(-\frac{\sqrt{3}}{8}\right) \]

\[ = \frac{\pi}{8} + \frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{64} \]

Simplify:

\[ = \frac{\pi}{8} + \frac{8\sqrt{3} - \sqrt{3}}{64} \]

\[ = \frac{\pi}{8} + \frac{7\sqrt{3}}{64} \]

Step 5: Calculate \(9a + 8b\)

From the result, we identify \(a = \frac{1}{8}\) and \(b = \frac{7}{64}\). The expression to evaluate is:

\[ 9a + 8b = 9 \times \frac{1}{8} + 8 \times \frac{7}{64} \]

\[ = \frac{9}{8} + \frac{56}{64} \]

\[ = \frac{9}{8} + \frac{7}{8} = 2 \]

The value of \(9a + 8b\) is \(2\). The final answer is 2.