If 0 3 cos 4 x ? d x = a + b 3 , where a and b are rational numbers, Then 9 a + 8 b is equal to: [2024]

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Solution

Q.
If $\int_{0}^{\frac{π}{3}} \cos^{4} x d x = a π + b \sqrt{3},$ where $a$ and $b$ are rational numbers, Then $9 a + 8 b$ is equal to: [2024]

1 2

2 1

3 3

4 $\frac{3}{2}$

Ans.
(1)

Let $I = \int_{0}^{π / 3} \cos^{4} x d x = \int_{0}^{π / 3} {(\frac{1 + \cos 2 x}{2})}^{2} d x$

$= \frac{1}{4} \int_{0}^{π / 3} (1 + \cos^{2} 2 x + 2 \cos 2 x) d x$

$= \frac{1}{4} \int_{0}^{π / 3} (1 + 2 \cos 2 x + (\frac{1 + \cos 4 x}{2})) d x$

$= \frac{1}{4} \int_{0}^{π / 3} (\frac{3}{2} + 2 \cos 2 x + \frac{\cos 4 x}{2}) d x$

$= \frac{1}{4} {[\frac{3}{2} x + \sin 2 x + \frac{\sin 4 x}{8}]}_{0}^{π / 3}$

$= \frac{1}{4} [\frac{π}{2} + \frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{16}] = \frac{π}{8} + \frac{7 \sqrt{3}}{64} \Rightarrow a = \frac{1}{8}, b = \frac{7}{64}$

$∴ 9 a + 8 b = \frac{9}{8} + \frac{7}{8} = 2$

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