Answer is: option3
\( \frac{2}{7}x^3\sqrt{x} - \frac{4}{3}x\sqrt{x} + C \)Solution:
\[ (x^2 - 2)\sqrt{x} = x^2\sqrt{x} - 2\sqrt{x} \]
Recall: \( \sqrt{x} = x^{1/2} \), so:
- \( x^2\sqrt{x} = x^{2 + 1/2} = x^{5/2} \)
- \( 2\sqrt{x} = 2x^{1/2} \)
So the integral becomes:
\[ \int (x^{5/2} - 2x^{1/2}) \, dx \]
\[ \int x^{5/2} \, dx = \frac{x^{7/2}}{7/2} = \frac{2}{7}x^{7/2} \]
\[ \int 2x^{1/2} \, dx = 2 \cdot \frac{x^{3/2}}{3/2} = \frac{4}{3}x^{3/2} \]
\[ \int (x^2 - 2)\sqrt{x} \, dx = \frac{2}{7}x^{7/2} - \frac{4}{3}x^{3/2} + C \]
- \( x^{7/2} = x^3\sqrt{x} \)
- \( x^{3/2} = x\sqrt{x} \)
So the final answer is: (C)
\[ \frac{2}{7}x^3\sqrt{x} - \frac{4}{3}x\sqrt{x} + C \]