Answer
$F(x) = \frac{x^{4.4}}{4.4} - \frac{2}{\sqrt 2}x^{\sqrt 2} + C$
Work Step by Step
$f(x) = x^{3.4} - 2x^{\sqrt 2 -1}$
$F(x) = \frac{x^{4.4}}{4.4} - 2(\frac{x^{\sqrt 2 -1+1}}{\sqrt 2 -1+1})$
$F(x) = \frac{x^{4.4}}{4.4} - 2(\frac{x^{\sqrt 2}}{\sqrt 2})$
$F(x) = \frac{x^{4.4}}{4.4} - \frac{2}{\sqrt 2}x^{\sqrt 2}$
Check:
$F(x) = \frac{1}{4.4}x^{4.4} - \frac{2}{\sqrt 2}x^{\sqrt 2}$
$F'(x) = \frac{4.4}{4.4}x^{4.4-1}- \frac{2\sqrt 2}{\sqrt 2}x^{\sqrt 2-1}$
$F'(x) = 1x^{3.4}- \frac{2}{1}x^{\sqrt 2-1}$
$F'(x) = x^{3.4}- 2x^{\sqrt 2-1}$