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Category: Integration

Evaluate-1-1-1-x-2-n-2-dx-for-n-Z-0-i-e-0-1-2-and-a-n-0-mod-2-b-n-1-mod-2-

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

e-t-ln-1-e-t-1-e-t-dt-

Question Number 17512 by tawa tawa last updated on 06/Jul/17 $$\int\:\frac{\mathrm{e}^{−\mathrm{t}} \:\mathrm{ln}\left(\mathrm{1}\:+\:\mathrm{e}^{−\mathrm{t}} \right)}{\mathrm{1}\:+\:\mathrm{e}^{−\mathrm{t}} }\:\mathrm{dt} \\ $$ Answered by sma3l2996 last updated on 07/Jul/17 $${I}=\int\frac{{e}^{−{t}} {ln}\left(\mathrm{1}+{e}^{−{t}}…

tan-1-x-1-x-1-dx-

Question Number 17506 by tawa tawa last updated on 06/Jul/17 $$\int\:\mathrm{tan}^{−\mathrm{1}} \left(\sqrt{\frac{\mathrm{x}\:+\:\mathrm{1}}{\mathrm{x}\:−\:\mathrm{1}}}\right)\:\mathrm{dx} \\ $$ 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}}} \\…