$\int \frac{(1 + x)^{3}}{\sqrt{x}} d x =$

Easy

A

$2 \sqrt{x} + 2 x^{3 / 2} + \frac{6}{5} x^{5 / 2} + \frac{2}{7} x^{7 / 2} + c$
B

$2 \sqrt{x} + 2 x^{3 / 2} + \frac{6}{5} x^{5 / 2} - \frac{2}{7} x^{7 / 2} + c$
C

$2 \sqrt{x} - 2 x^{3 / 2} + \frac{6}{5} x^{5 / 2} + \frac{2}{7} x^{7 / 2} + c$
D

$2 \sqrt{x} + 2 x^{3 / 2} - \frac{6}{5} x^{5 / 2} + \frac{2}{7} x^{7 / 2} + c$

Solution

$\begin{array}{l} \int \frac{(1 + x)^{3}}{\sqrt{x}} d x \\ = \int \frac{1 + 3 x + 3 x^{2} + x^{3}}{\sqrt{x}} d x \\ = \int x^{- \frac{1}{2}} d x + 3 \int x^{\frac{1}{2}} d x + 3 \int x^{\frac{3}{2}} d x + \int x^{\frac{5}{2}} d x \\ \begin{array}{l} = \frac{x^{\frac{1}{2}}}{\frac{1}{2}} + 3 \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + 3 \cdot \frac{x^{\frac{5}{2}}}{\frac{5}{2}} + \frac{x^{\frac{7}{2}}}{\frac{7}{2}} + c \\ = 2 \sqrt{x} + 2 x^{\frac{3}{2}} + \frac{6}{5} x^{\frac{5}{2}} + \frac{2}{7} x^{\frac{7}{2}} + c \end{array} \end{array}$

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