Integral of \( \displaystyle 2 \sqrt{2} x^{\frac{3}{2}} - \frac{\sqrt{2}}{2 \sqrt{x}} \)

Problem 4.477 · hard

1. \[ \int \left(2 \sqrt{2} x^{\frac{3}{2}} - \frac{\sqrt{2}}{2 \sqrt{x}}\right)\, dx \]
   integralStart with the integral of the given expression.✓ Proved
2. \[ = - \int \frac{\sqrt{2}}{2 \sqrt{x}}\, dx + \int 2 \sqrt{2} x^{\frac{3}{2}}\, dx \]
   linearity rewriteSplit the integral into two parts. Rewrite the second term using a fractional exponent.✓ Proved
3. \[ = - \frac{\sqrt{2} \int \frac{1}{\sqrt{x}}\, dx}{2} + 2 \sqrt{2} \int x^{\frac{3}{2}}\, dx \]
   linearityPull out the constant coefficients.✓ Proved
4. \[ = \frac{4 \sqrt{2} x^{\frac{5}{2}}}{5} - \sqrt{2} \sqrt{x} \]
   antiderivative simplifyApply the power rule for integration. Simplify the coefficients.✓ Proved