Answer
$$\frac{{\sqrt 2 {x^{\sqrt 2 }}}}{2} + C $$
Work Step by Step
$$\eqalign{
& \int {{x^{\sqrt 2 - 1}}dx} \cr
& {\text{integrate using the power rule }}\int {{x^n}dx = \frac{{{x^{n + 1}}}}{{n + 1}} + C,{\text{ }}where{\text{ }}n \ne - 1.{\text{ }}S{\text{o}}{\text{,}}} \cr
& = \frac{{{x^{\sqrt 2 - 1 + 1}}}}{{\sqrt 2 - 1 + 1}} + C \cr
& {\text{simplifying}} \cr
& = \frac{{{x^{\sqrt 2 }}}}{{\sqrt 2 }} + C \cr
& {\text{rationalizing the denominator:}} \cr
& = \frac{{\sqrt 2 {x^{\sqrt 2 }}}}{{\sqrt 2 \sqrt 2 }} + C \cr
& = \frac{{\sqrt 2 {x^{\sqrt 2 }}}}{2} + C \cr} $$
