Question Number 40134 by maxmathsup by imad last updated on 16/Jul/18
$${calculate}\:\:\int_{\mathrm{1}} ^{\mathrm{2}} \:\:\:\frac{{x}^{\mathrm{3}} }{\left(\mathrm{1}+{x}^{\mathrm{4}} \right)^{\mathrm{2}} }{dx} \\ $$
Commented by maxmathsup by imad last updated on 18/Jul/18
![I = −(1/4) ∫_1 ^2 ((−4x^3 )/((1+x^4 )^2 ))dx=−(1/4)[ (1/(1+x^4 ))]_1 ^2 =−(1/4){ (1/(17)) −(1/2)} ⇒ I =(1/8) −(1/(68))](https://www.tinkutara.com/question/Q40267.png)
$${I}\:=\:−\frac{\mathrm{1}}{\mathrm{4}}\:\int_{\mathrm{1}} ^{\mathrm{2}} \:\:\frac{−\mathrm{4}{x}^{\mathrm{3}} }{\left(\mathrm{1}+{x}^{\mathrm{4}} \right)^{\mathrm{2}} }{dx}=−\frac{\mathrm{1}}{\mathrm{4}}\left[\:\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{4}} }\right]_{\mathrm{1}} ^{\mathrm{2}} \\ $$$$=−\frac{\mathrm{1}}{\mathrm{4}}\left\{\:\:\frac{\mathrm{1}}{\mathrm{17}}\:−\frac{\mathrm{1}}{\mathrm{2}}\right\}\:\Rightarrow \\ $$$${I}\:=\frac{\mathrm{1}}{\mathrm{8}}\:−\frac{\mathrm{1}}{\mathrm{68}} \\ $$