If integral then find the value of .

JEE MainMathematicsModerateDefinite Integration

If
integral $\int_{2}^{4}\mspace{2mu}\sqrt{\frac{x^{2} - 4}{x^{4}}}dx = ln(2 + \sqrt{3}) - \frac{\lambda\sqrt{3}}{64},$ then
find the value of $\lambda$ . $\$

Show answer and solution

\int_{2}^{4}\mspace{2mu}\sqrt{\frac{x^{2} - 4}{x^{4}}}

dx Put x = 2

\sec\theta

dx =

2\sec\theta\tan\theta d\theta

= \int_{0}^{\pi/3}\mspace{2mu}\frac{2tan\theta}{4\sec^{2}\theta} \cdot 2\sec\theta\tan\theta d\theta

= \int_{0}^{\frac{\pi}{3}}\mspace{2mu}\frac{\sin^{2}\theta}{\cos\theta}d\theta = \int_{0}^{\frac{\pi}{3}}\mspace{2mu}\left( \sec\theta - \cos\theta \right)d\theta

\mathcal{= l}n(sec\theta + tan\theta)_{0}^{\pi/3} - (sin\theta)_{0}^{\pi/3}

= ln(2 + \sqrt{3}) - \frac{\sqrt{3}}{2}

Source: Grass-G / M-13 Mathematics Paper 4 Nov.docx

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