Problem Set: Integrals Resulting in Inverse Trigonometric

In the following exercises (1-6), evaluate each integral in terms of an inverse trigonometric function.

1. ${\displaystyle\int }_{0}^{\sqrt{3}\text{/}2}\frac{dx}{\sqrt{1-{x}^{2}}}$

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${ \sin }^{-1}x{|}_{0}^{\sqrt{3}\text{/}2}=\frac{\pi }{3}$

2. ${\displaystyle\int }_{-1\text{/}2}^{1\text{/}2}\frac{dx}{\sqrt{1-{x}^{2}}}$

3. ${\displaystyle\int }_{\sqrt{3}}^{1}\frac{dx}{\sqrt{1+{x}^{2}}}$

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${ \tan }^{-1}x{|}_{\sqrt{3}}^{1}=-\frac{\pi }{12}$

4. ${\displaystyle\int }_{1\text{/}\sqrt{3}}^{\sqrt{3}}\frac{dx}{1+{x}^{2}}$

5. ${\displaystyle\int }_{1}^{\sqrt{2}}\frac{dx}{|x|\sqrt{{x}^{2}-1}}$

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${ \sec }^{-1}x{|}_{1}^{\sqrt{2}}=\frac{\pi }{4}$

6. ${\displaystyle\int }_{1}^{2\text{/}\sqrt{3}}\frac{dx}{|x|\sqrt{{x}^{2}-1}}$