Category: Integration

Question Number 17525 by alex041103 last updated on 07/Jul/17 $$\mathrm{Evaluate}\underset{-1}{\stackrel{1}{\int }}{(1-x^2)}^{\frac{n}{2}}dx\mathrm{for}$$$$n\in \mathbb{Z}\cap [0;\mathrm{\infty })(\mathrm{i}.\mathrm{e}.0,1,2,\dots )\mathrm{and}:$$$$\mathbf{a})n\equiv 0\left(mod2\right)$$$$\mathit{b})n\equiv 1\left(mod2\right)$$ Commented by alex041103…

Question Number 17514 by sma3l2996 last updated on 07/Jul/17 $$S\left(n\right)={\int }_0^1{x}^{2n}sin\left(2n\pi x\right)dx$$ Commented by alex041103 last updated on 07/Jul/17 $$\mathrm{Is}n\mathrm{a}\mathrm{whole}\mathrm{number}?$$…

Question Number 17506 by tawa tawa last updated on 06/Jul/17 $$\int {\tan }^{-1}\left(\sqrt{\frac{\mathrm{x}+1}{\mathrm{x}-1}}\right)\mathrm{dx}$$ Answered by sma3l2996 last updated on 06/Jul/17 $${u}={tan}^{−\mathrm{1}} \left(\sqrt{\frac{{x}+\mathrm{1}}{{x}−\mathrm{1}}}\right)\Rightarrow{u}'=\frac{−\mathrm{1}}{\mathrm{2}{x}\sqrt{\left({x}−\mathrm{1}\right)\left({x}+\mathrm{1}\right)}}=\frac{−\mathrm{1}}{\mathrm{2}{x}\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}} \…

Question Number 148570 by mathmax by abdo last updated on 29/Jul/21 $$\mathrm{calculate}{\int }_0^{\mathrm{\infty }}\frac{\mathrm{logx}}{{\mathrm{x}}^2+\mathrm{x}+1}\mathrm{dx}$$ Answered by Ar Brandon last updated on 29/Jul/21…

Question Number 148566 by mathmax by abdo last updated on 29/Jul/21 $$\mathrm{calculate}{\mathrm{U}}_{\mathrm{n}}=\int {\int }_{[\frac{1}{\mathrm{n}},\mathrm{n}[}\frac{\cos ({\mathrm{x}}^2+{\mathrm{y}}^2)}{{\mathrm{x}}^2+{\mathrm{y}}^2}\mathrm{dxdy}$$$$\mathrm{and}\mathrm{determine}{\lim }_{\mathrm{n}\to +\mathrm{\infty }}{\mathrm{U}}_{\mathrm{n}}$$$$\mathrm{nature}\:\mathrm{of}\:\Sigma\:\mathrm{U}_{\mathrm{n}} ? \…