Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

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

∫x^7 (√(1−x^3 ))dx=?

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

Commented by TITA last updated on 11/Dec/20

please help

$${please}\:{help} \\ $$

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

x^3 =t ⇒x=t^(1/3)  ⇒∫ x^7 (√(1−x^3 ))dx =∫ t^(7/3) (1−t)^(1/2) ×(1/3)t^((1/3)−1)  dt  =(1/3)∫  t^((8/3)−1) (1−t)^((3/2)−1)  dt =(1/3)B_(inc) ((8/3),(3/2))  B_(inc)   means incomplete B

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

Terms of Service

Privacy Policy

Contact: info@tinkutara.com