Integration Questions

Question Number 4957 by ankit chakravarti last updated on 26/Mar/16

$$\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{4}{x}^{\mathrm{3}} \left\{\frac{{d}^{\mathrm{2}} }{{dx}^{\mathrm{2}} }\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{5}} \right\}{dx}=? \\$$

Commented byankit chakravarti last updated on 26/Mar/16

$${please}\:{solve}..... \\$$

Answered by Yozzii last updated on 26/Mar/16

$${Integrating}\:{by}\:{parts}\:{we}\:{obtain} \\$$ $${I}=\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{4}{x}^{\mathrm{3}} \left\{\frac{{d}^{\mathrm{2}} }{{dx}^{\mathrm{2}} }\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{5}} \right\}{dx} \\$$ $${I}=\mathrm{4}{x}^{\mathrm{3}} \frac{{d}}{{dx}}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{5}} \mid_{\mathrm{0}} ^{\mathrm{1}} −\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{12}{x}^{\mathrm{2}} \left(\frac{{d}}{{dx}}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{5}} \right){dx} \\$$ $${I}=\mathrm{0}−\mathrm{12}\left(\left[{x}^{\mathrm{2}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{5}} \right]_{\mathrm{0}} ^{\mathrm{1}} −\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{2}{x}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{5}} {dx}\right) \\$$ $${I}=\left(−\mathrm{12}\right)\int_{\mathrm{0}} ^{\mathrm{1}} \left(−\mathrm{2}{x}\right)\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{5}} {dx} \\$$ $${Let}\:{u}=\mathrm{1}−{x}^{\mathrm{2}} \Rightarrow{du}=−\mathrm{2}{xdx} \\$$ $$\Rightarrow{I}=−\mathrm{12}\int_{\mathrm{1}} ^{\mathrm{0}} {u}^{\mathrm{5}} {du}=\mathrm{12}\int_{\mathrm{0}} ^{\mathrm{1}} {u}^{\mathrm{5}} {du}=\frac{\mathrm{12}}{\mathrm{6}}{u}^{\mathrm{6}} \mid_{\mathrm{0}} ^{\mathrm{1}} =\mathrm{2}. \\$$ $$\\$$