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Question Number 125542 by TITA last updated on 11/Dec/20

$$\int x^7\sqrt{1-x^3}dx=?$$

Commented by TITA last updated on 11/Dec/20

$$pleasehelp$$

Answered by mathmax by abdo last updated on 12/Dec/20

$${\mathrm{x}}^3=\mathrm{t}\Rightarrow \mathrm{x}={\mathrm{t}}^{\frac{1}{3}}\Rightarrow \int {\mathrm{x}}^7\sqrt{1-{\mathrm{x}}^3}\mathrm{dx}=\int {\mathrm{t}}^{\frac{7}{3}}{(1-\mathrm{t})}^{\frac{1}{2}}\times \frac{1}{3}{\mathrm{t}}^{\frac{1}{3}-1}\mathrm{dt}$$$$=\frac{1}{3}\int {\mathrm{t}}^{\frac{8}{3}-1}{(1-\mathrm{t})}^{\frac{3}{2}-1}\mathrm{dt}=\frac{1}{3}{\mathrm{B}}_{\mathrm{inc}}(\frac{8}{3},\frac{3}{2})$$$${\mathrm{B}}_{\mathrm{inc}}\mathrm{means}\mathrm{incomplete}\mathrm{B}$$$$$$

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