Question

\[\int\frac{\left( 1 + x \right)^3}{\sqrt{x}} dx\] 

Answer 1

\[\int\frac{\left( 1 + x \right)^3}{\sqrt{x}}dx\]
\[ = \int\left( \frac{1 + x^3 + 3 \left( 1 \right)^2 x + 3\left( 1 \right) x^2}{\sqrt{x}} \right)dx\]
\[ = \int\left( \frac{1 + x^3 + 3x + 3 x^2}{\sqrt{x}} \right) dx\]
\[ = \int\left( \frac{1}{\sqrt{x}} + \frac{x^3}{\sqrt{x}} + \frac{3x}{\sqrt{x}} + \frac{3 x^2}{\sqrt{x}} \right)dx\]
\[ = \int\left( x^{- \frac{1}{2}} + x^\frac{5}{2} + 3 x^\frac{1}{2} + 3 x^\frac{3}{2} \right)dx\]
\[ = \left[ \frac{x^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1} + \frac{x^\frac{5}{2} + 1}{\frac{5}{2} + 1} + 3\frac{x^\frac{1}{2} + 1}{\frac{1}{2} + 1} + 3\frac{x^\frac{3}{2} + 1}{\frac{3}{2} + 1} \right] + C\]
\[ = 2\sqrt{x} + \frac{2}{7} x^\frac{7}{2} + 2 x^\frac{3}{2} + \frac{6}{5} x^\frac{5}{2} + C\]