#### Answer

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

#### Work Step by Step

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