Integral of x2\frac{\sqrt{x}}{2}

The calculator will find the integral/antiderivative of x2\frac{\sqrt{x}}{2}, with steps shown.

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Your Input

Find x2dx\int \frac{\sqrt{x}}{2}\, dx.

Solution

Apply the constant multiple rule cf(x)dx=cf(x)dx\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx with c=12c=\frac{1}{2} and f(x)=xf{\left(x \right)} = \sqrt{x}:

x2dx=(xdx2){\color{red}{\int{\frac{\sqrt{x}}{2} d x}}} = {\color{red}{\left(\frac{\int{\sqrt{x} d x}}{2}\right)}}

Apply the power rule xndx=xn+1n+1\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1} (n1)\left(n \neq -1 \right) with n=12n=\frac{1}{2}:

xdx2=x12dx2=x12+112+12=(2x323)2\frac{{\color{red}{\int{\sqrt{x} d x}}}}{2}=\frac{{\color{red}{\int{x^{\frac{1}{2}} d x}}}}{2}=\frac{{\color{red}{\frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}}}}{2}=\frac{{\color{red}{\left(\frac{2 x^{\frac{3}{2}}}{3}\right)}}}{2}

Therefore,

x2dx=x323\int{\frac{\sqrt{x}}{2} d x} = \frac{x^{\frac{3}{2}}}{3}

Add the constant of integration:

x2dx=x323+C\int{\frac{\sqrt{x}}{2} d x} = \frac{x^{\frac{3}{2}}}{3}+C

Answer

x2dx=x323+C\int \frac{\sqrt{x}}{2}\, dx = \frac{x^{\frac{3}{2}}}{3} + CA