AP Calculus AB / Integration and Accumulation of Change / Question No. 4

4. $\int (x^{2} - 2) \sqrt{x} d x =$

\frac{2}{5} x^{2} \sqrt{x} - \frac{2}{3} x \sqrt{x} + C

\frac{2}{5} x^{2} \sqrt{x} - \frac{4}{3} x \sqrt{x} + C

\frac{2}{7} x^{3} \sqrt{x} - \frac{4}{3} x \sqrt{x} + C

\frac{2}{7} x^{3} \sqrt{x} - \frac{2}{3} x^{2} \sqrt{x} + C

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:

So the integral becomes:

$\int (x^{5 / 2} - 2 x^{1 / 2}) d x$

$\int x^{5 / 2} d x = \frac{x^{7 / 2}}{7 / 2} = \frac{2}{7} x^{7 / 2}$

$\int 2 x^{1 / 2} d x = 2 \cdot \frac{x^{3 / 2}}{3 / 2} = \frac{4}{3} x^{3 / 2}$

$\int (x^{2} - 2) \sqrt{x} d x = \frac{2}{7} x^{7 / 2} - \frac{4}{3} x^{3 / 2} + C$

So the final answer is: (C)

$\frac{2}{7} x^{3} \sqrt{x} - \frac{4}{3} x \sqrt{x} + C$

Previous Next