If 0 5 cos x ( 1 + cos x cos 3 x + cos 2 x + cos 3 x cos 3 x ) d x 1 + 5 cos x = k 16 , then k is equal to ______. [2023]

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Solution

Q.
If $\int_{0}^{π} \frac{5^{\cos x} (1 + \cos x \cos 3 x + \cos^{2} x + \cos^{3} x \cos 3 x) d x}{1 + 5^{\cos x}} = \frac{k π}{16},$ then $k$ is equal to ______. [2023]

Ans.
(13)

$Let$

$I = \int_{0}^{π} \frac{5^{\cos x} (1 + \cos x \cdot \cos 3 x + \cos^{2} x + \cos^{3} x \cdot \cos 3 x)}{1 + 5^{\cos x}} d x$ ...(i)

$I = \int_{0}^{π} \frac{5^{\cos (π - x)} (1 + \cos (π - x) \cdot \cos 3 (π - x) + \cos^{2} (π - x) + \cos^{3} (π - x) \cdot \cos 3 (π - x))}{1 + 5^{\cos (π - x)}} d x$

$\Rightarrow I = \int_{0}^{π} \frac{5^{- \cos x} (1 + \cos x \cdot \cos 3 x + \cos^{2} x + \cos^{3} x \cdot \cos 3 x) d x}{1 + 5^{- \cos x}}$ ...(ii)

$Adding (i) and (ii), we get$

$2 I = \int_{0}^{π} (1 + \cos x \cdot \cos 3 x + \cos^{2} x + \cos^{3} x \cdot \cos 3 x) d x$

$\Rightarrow 2 I = 2 \int_{0}^{π / 2} (1 + \cos x \cdot \cos 3 x + \cos^{2} x + \cos^{3} x \cdot \cos 3 x) d x$ ...(iii)

$\Rightarrow I = \int_{0}^{π / 2} (1 + \sin x (- \sin 3 x) + \sin^{2} x - \sin^{3} x \cdot \sin 3 x) d x$ ...(iv)

$Adding (iii) and (iv), we get$

$\Rightarrow 2 I = \int_{0}^{π / 2} (3 + \cos 4 x + \cos^{3} x \cdot \cos 3 x - \sin^{3} x \cdot \sin 3 x) d x$

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

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

$\Rightarrow 2 I = \frac{13}{4} \times \frac{π}{2} + \frac{7}{4} {(\frac{\sin 4 x}{4})}_{0}^{π / 2} \Rightarrow I = \frac{13 π}{16}$

$Hence, k = 13$

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