Question

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

Answer 1

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