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

Easy

A

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

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

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

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

Solution

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

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