Show that $\displaystyle\int_{1}^{2}{\sqrt{1-\frac{1}{{{x}^{2}}}}}\mathrm{d}x=\sqrt{3}-\frac{\pi }{3

Question

Show that $\displaystyle\int_{1}^{2}{\sqrt{1-\frac{1}{{{x}^{2}}}}}\mathrm{d}x=\sqrt{3}-\frac{\pi }{3}$ by using the substitution $\frac{1}{x}=\cos 	heta $.

Accepted Answer

$\sqrt{3} - \frac{\pi}{3}$

Evaluation of a Definite Integral Using Trigonometric Substitution

Evaluation of a Definite Integral Using Trigonometric Substitution

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