# Chapter 4 - Applications of the Derivative - 4.9 Antiderivatives - 4.9 Exercises - Page 238: 54

$${\sin ^{ - 1}}\frac{x}{7} + C$$

#### Work Step by Step

\eqalign{ & \int {{{\left( {49 - {x^2}} \right)}^{ - 1/2}}dx} \cr & {\text{write the negative fractional exponent as a radical}} \cr & = \int {\frac{1}{{\sqrt {49 - {x^2}} }}dx} \cr & {\text{from the table 4}}{\text{.10 }}\int {\frac{{dx}}{{\sqrt {{a^2} - {x^2}} }}} = {\sin ^{ - 1}}\frac{x}{a} + C \cr & {\text{letting }}a = 7 \cr & = {\sin ^{ - 1}}\frac{x}{7} + C \cr & {\text{check by differentiation}} \cr & {\text{ = }}\frac{d}{{dx}}\left( {{{\sin }^{ - 1}}\frac{x}{7} + C} \right) \cr & {\text{ = }}\frac{d}{{dx}}\left( {{{\sin }^{ - 1}}\frac{x}{7}} \right) + \frac{d}{{dx}}\left( C \right) \cr & {\text{ = }}\left( {\frac{1}{{\sqrt {{{\left( 7 \right)}^2} - {x^2}} }}} \right) + 0 \cr & = \frac{1}{{\sqrt {49 - {x^2}} }} \cr}

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