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Category: Integration

nice-calculus-evaluation-of-0-xe-x-1-e-x-dx-solution-1-e-x-t-x-ln-1-t-e-x-dx-dt-0-

Question Number 135525 by mnjuly1970 last updated on 13/Mar/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:….\:{nice}\:…………….\:{calculus}… \\ $$$$\:\:\:\:\:{evaluation}\:{of}\:::\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\infty} {xe}^{−{x}} \sqrt{\mathrm{1}−{e}^{−{x}} }\:{dx} \\ $$$$\:\:\:\:{solution}::\: \\ $$$$\:\:\:\:\mathrm{1}−{e}^{−{x}} ={t}\:\:\Rightarrow\:\left\{_{\:{x}=−{ln}\left(\mathrm{1}−{t}\right)} ^{\:{e}^{−{x}} {dx}={dt}} \right. \\…

Question-135513

Question Number 135513 by Gaurav500 last updated on 13/Mar/21 Answered by MJS_new last updated on 13/Mar/21 $$\int\frac{{dx}}{\:\sqrt{{x}}+\sqrt{{x}+\mathrm{1}}+\sqrt{{x}+\mathrm{2}}}= \\ $$$$\:\:\:\:\:\left[{t}={x}+\mathrm{1}\:\rightarrow\:{dx}={dt}\right] \\ $$$$=\int\frac{{dt}}{\:\sqrt{{t}−\mathrm{1}}+\sqrt{{t}}+\sqrt{{t}+\mathrm{1}}}=\underset{{k}=\mathrm{1}} {\overset{\mathrm{6}} {\sum}}{I}_{{k}} \:+{Ci} \\…

hi-guyz-let-s-try-this-I-0-1-sin-2-x-cos-3-x-dx-

Question Number 135495 by greg_ed last updated on 13/Mar/21 $$\boldsymbol{\mathrm{hi}},\:\boldsymbol{\mathrm{guyz}}\:! \\ $$$$\boldsymbol{\mathrm{let}}'\boldsymbol{\mathrm{s}}\:\boldsymbol{\mathrm{try}}\:\boldsymbol{\mathrm{this}}\::\:\boldsymbol{\mathrm{I}}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\boldsymbol{{sin}}^{\mathrm{2}} \boldsymbol{{x}}}{\boldsymbol{{cos}}^{\mathrm{3}} \boldsymbol{{x}}}\boldsymbol{{dx}}. \\ $$ Answered by mathmax by abdo last updated…

for-f-x-dx-F-x-c-and-sgn-x-x-x-x-x-x-0-let-sgn-x-0-for-x-0-does-sgn-f-x-f-x-dx-sgn-f-x-f-x-dx-sgn-f-x-is-just-a-constant-1-or-0-

Question Number 4344 by Filup last updated on 12/Jan/16 $$\mathrm{for}\:\int{f}\left({x}\right){dx}={F}\left({x}\right)+{c} \\ $$$$\mathrm{and}\:{sgn}\left({x}\right)=\frac{{x}}{\mid{x}\mid}=\frac{\mid{x}\mid}{{x}}\:\:\:\forall{x}\neq\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{let}\:\mathrm{sgn}\left({x}\right)=\mathrm{0}\:\mathrm{for}\:{x}=\mathrm{0} \\ $$$$ \\ $$$$\mathrm{does}\: \\ $$$$\int{sgn}\left({f}\left({x}\right)\right){f}\left({x}\right){dx}={sgn}\left({f}\left({x}\right)\right)\int{f}\left({x}\right){dx} \\ $$$$\because{sgn}\left({f}\left({x}\right)\right)\:\mathrm{is}\:\mathrm{just}\:\mathrm{a}\:\mathrm{constant}\:\pm\mathrm{1}\:\mathrm{or}\:\mathrm{0}. \\ $$ Commented…