Chapter 4: Applications of Derivatives - Section 4.7 - Antiderivatives - Exercises 4.7 - Page 239: 50

$$\frac{{\sqrt 2 {x^{\sqrt 2 }}}}{2} + C $$

$$\eqalign{ & \int {{x^{\sqrt 2 - 1}}} dx \cr & \sqrt 2 - 1{\text{ is a constant}}{\text{, then using }}\int {{x^a}} dx = \frac{{{x^{a + 1}}}}{{a + 1}} + C \cr & = \frac{{{x^{\sqrt 2 - 1 + 1}}}}{{\sqrt 2 - 1 + 1}} + C \cr & {\text{simplifying, we get:}} \cr & = \frac{{{x^{\sqrt 2 }}}}{{\sqrt 2 }} + C \cr & = \frac{{\sqrt 2 {x^{\sqrt 2 }}}}{{\sqrt 2 \sqrt 2 }} + C \cr & = \frac{{\sqrt 2 {x^{\sqrt 2 }}}}{2} + C \cr} $$