$\int x^{\frac{1}{3}} {(2 + x^{\frac{1}{2}})}^{2} d x is equal to$

Moderate

A

$3 x^{\frac{4}{3}} + \frac{24}{11} x^{\frac{11}{6}} + \frac{3}{7} x^{\frac{7}{3}} + c$
B

$3 x^{\frac{3}{4}} + \frac{24}{11} x^{\frac{11}{6}} + \frac{3}{7} x^{\frac{7}{3}} + c$
C

$3 x^{\frac{3}{4}} + \frac{24}{11} x^{\frac{11}{6}} + \frac{3}{7} x^{\frac{3}{7}} + c$
D

$3 x^{\frac{3}{4}} + \frac{24}{11} x^{\frac{11}{6}} + \frac{7}{3} x^{\frac{7}{3}} + c$

Solution

$\int x^{\frac{1}{3}} {(2 + x^{\frac{1}{2}})}^{2} d x$
$Let x = t^{6}$
$\begin{array}{l} d x = 6 t^{5} d t \\ = \int t^{2} {(2 + t^{3})}^{2} 6 t^{5} d t \\ = 6 \int t^{7} (4 + t^{6} + 4 t^{3}) d t \\ = 6 \int (4 t^{7} + 4 t^{10} + t^{13}) d t \\ = 6 \int (4 \frac{t^{8}}{8} + 4 \frac{t^{11}}{11} + \frac{t^{14}}{14}) + C \\ = 3 x^{\frac{4}{3}} + \frac{24}{11} x^{\frac{11}{6}} + \frac{3}{7} \cdot x^{\frac{7}{3}} + C \end{array}$

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