# Solve for: integral of x^3sqrt(1-x^2) x

## Expression: $\int{ {x}^{3}\sqrt{ 1-{x}^{2} } } \mathrm{d} x$

Use the substitution $t=1-{x}^{2}$ to transform the integral

$\int{ \frac{ t\sqrt{ t }-\sqrt{ t } }{ 2 } } \mathrm{d} t$

Use the property of integral $\begin{array} { l }\int{ a \times f\left( x \right) } \mathrm{d} x=a \times \int{ f\left( x \right) } \mathrm{d} x,& a \in ℝ\end{array}$

$\frac{ 1 }{ 2 } \times \int{ t\sqrt{ t }-\sqrt{ t } } \mathrm{d} t$

Use $\sqrt[n]{{a}^{m}}={a}^{\frac{ m }{ n }}$ to transform the expression

$\frac{ 1 }{ 2 } \times \int{ t \times {t}^{\frac{ 1 }{ 2 }}-\sqrt{ t } } \mathrm{d} t$

Use $\sqrt[n]{{a}^{m}}={a}^{\frac{ m }{ n }}$ to transform the expression

$\frac{ 1 }{ 2 } \times \int{ t \times {t}^{\frac{ 1 }{ 2 }}-{t}^{\frac{ 1 }{ 2 }} } \mathrm{d} t$

Calculate the product

$\frac{ 1 }{ 2 } \times \int{ {t}^{\frac{ 3 }{ 2 }}-{t}^{\frac{ 1 }{ 2 }} } \mathrm{d} t$

Use the property of integral $\int{ f\left( x \right)\pmg\left( x \right) } \mathrm{d} x=\int{ f\left( x \right) } \mathrm{d} x\pm\int{ g\left( x \right) } \mathrm{d} x$

$\frac{ 1 }{ 2 } \times \left( \int{ {t}^{\frac{ 3 }{ 2 }} } \mathrm{d} t-\int{ {t}^{\frac{ 1 }{ 2 }} } \mathrm{d} t \right)$

Use $\begin{array} { l }\int{ {x}^{n} } \mathrm{d} x=\frac{ {x}^{n+1} }{ n+1 },& n≠-1\end{array}$ to evaluate the integral

$\frac{ 1 }{ 2 } \times \left( \frac{ 2{t}^{2}\sqrt{ t } }{ 5 }-\int{ {t}^{\frac{ 1 }{ 2 }} } \mathrm{d} t \right)$

Use $\begin{array} { l }\int{ {x}^{n} } \mathrm{d} x=\frac{ {x}^{n+1} }{ n+1 },& n≠-1\end{array}$ to evaluate the integral

$\frac{ 1 }{ 2 } \times \left( \frac{ 2{t}^{2}\sqrt{ t } }{ 5 }-\frac{ 2t\sqrt{ t } }{ 3 } \right)$

Substitute back $t=1-{x}^{2}$

$\frac{ 1 }{ 2 } \times \left( \frac{ 2{\left( 1-{x}^{2} \right)}^{2} \times \sqrt{ 1-{x}^{2} } }{ 5 }-\frac{ 2\left( 1-{x}^{2} \right)\sqrt{ 1-{x}^{2} } }{ 3 } \right)$

Simplify the expression

$\frac{ \sqrt{ 1-{x}^{2} }\left( 1-2{x}^{2}+{x}^{4} \right) }{ 5 }-\frac{ \left( 1-{x}^{2} \right)\sqrt{ 1-{x}^{2} } }{ 3 }$

Add the constant of integration $C \in ℝ$

$\begin{array} { l }\frac{ \sqrt{ 1-{x}^{2} }\left( 1-2{x}^{2}+{x}^{4} \right) }{ 5 }-\frac{ \left( 1-{x}^{2} \right)\sqrt{ 1-{x}^{2} } }{ 3 }+C,& C \in ℝ\end{array}$

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