Category: Integration

Question Number 197376 by megrex last updated on 15/Sep/23 $$Doesanyoneknowhowtoprovethis?$$$$\int \int {\int }_V\frac{dxdydz}{1+x^4+y^4+z^4}=\frac{{\mathrm{\Gamma }}^4\left(\frac{1}{4}\right)}{4^4}$$$$whereVistheunitcube{[0,1]}^3$$$$Thankyou.$$$$ \…

Question Number 197292 by universe last updated on 12/Sep/23 $$\underset{n\to \mathrm{\infty }}{\lim }{\int }_0^1\frac{{nx}^{n-1}}{1+x}dx=?$$ Answered by witcher3 last updated on 12/Sep/23 $$\mathrm{2x}\leqslant\mathrm{1}+\mathrm{x}\leqslant\mathrm{2},\forall\mathrm{x}\in\left[\mathrm{0},\mathrm{1}\right] \…

Question Number 197212 by MathematicalUser2357 last updated on 10/Sep/23 $$\mathrm{Is}\int f\left(x\right)dx={\int }_0^x\underset{x\to t}{\lim }f\left(x\right)dt?$$ Commented by mahdipoor last updated on 10/Sep/23 $$iflimf\left(x\right),x\to t=f\left(t\right)$$$$\int{f}\left({x}\right){dx}=\int_{\mathrm{0}}…

Question Number 197191 by tri26112004 last updated on 10/Sep/23 $$\int \frac{1}{x^3-3x+7}dx$$ Answered by Frix last updated on 10/Sep/23 $$x^3-3x+7=(x-a)(x^2+ax+b)$$$${a}=−\frac{\sqrt[{\mathrm{3}}]{\mathrm{7}+\mathrm{3}\sqrt{\mathrm{5}}}+\sqrt[{\mathrm{3}}]{\mathrm{7}−\mathrm{3}\sqrt{\mathrm{5}}}}{\:\sqrt[{\mathrm{3}}]{\mathrm{2}}};\:{b}=−\frac{\mathrm{7}}{{a}}…

Question Number 197111 by cortano12 last updated on 08/Sep/23 $$\cancel{}$$ Answered by Frix last updated on 08/Sep/23 $$\int\frac{{dx}}{\:\sqrt{\mathrm{cos}^{\mathrm{3}} \:{x}\:\mathrm{sin}^{\mathrm{5}} \:{x}}}\:\overset{{t}=\mathrm{tan}\:{x}} {=} \…