Question Number 102127 by Study last updated on 06/Jul/20
$$\int_{\mathrm{0}} ^{\mathrm{2}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{3}} {dx}=? \\ $$$$\:{and}\:{write}\:{the}\:{furmollah} \\ $$
Answered by abdomathmax last updated on 06/Jul/20
![∫_0 ^2 (1−x^2 )^3 dx =−∫_0 ^2 (x^2 −1)^3 dx =−∫_0 ^2 (x^(6 ) +3x^4 (1)−3x^2 (1)^2 −1)dx =[−(x^7 /7) −(3/5)x^5 +x^3 +x]_0 ^2 −(2^7 /7)−(3/5)2^5 +2^3 +2 =....](https://www.tinkutara.com/question/Q102131.png)
$$\int_{\mathrm{0}} ^{\mathrm{2}} \left(\mathrm{1}−\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{3}} \:\mathrm{dx}\:=−\int_{\mathrm{0}} ^{\mathrm{2}} \left(\mathrm{x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{3}} \:\mathrm{dx} \\ $$$$=−\int_{\mathrm{0}} ^{\mathrm{2}} \left(\mathrm{x}^{\mathrm{6}\:} +\mathrm{3}{x}^{\mathrm{4}} \left(\mathrm{1}\right)−\mathrm{3}{x}^{\mathrm{2}} \left(\mathrm{1}\right)^{\mathrm{2}} −\mathrm{1}\right){dx} \\ $$$$=\left[−\frac{{x}^{\mathrm{7}} }{\mathrm{7}}\:−\frac{\mathrm{3}}{\mathrm{5}}{x}^{\mathrm{5}} \:+{x}^{\mathrm{3}} +{x}\right]_{\mathrm{0}} ^{\mathrm{2}} \\ $$$$−\frac{\mathrm{2}^{\mathrm{7}} }{\mathrm{7}}−\frac{\mathrm{3}}{\mathrm{5}}\mathrm{2}^{\mathrm{5}} \:+\mathrm{2}^{\mathrm{3}} \:+\mathrm{2}\:=…. \\ $$