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Category: Integration

How-do-you-all-to-prove-Integral-Prove-the-In-x-2-x-dx-1-3-In-x-3-

Question Number 164120 by Zaynal last updated on 14/Jan/22 $$\mathrm{How}\:\mathrm{do}\:\mathrm{you}\:\mathrm{all}\:\mathrm{to}\:\mathrm{prove}\:\mathrm{Integral}; \\ $$$$\:\boldsymbol{{Prove}}\:\boldsymbol{{the}}; \\ $$$$\:\int\:\frac{\left(\boldsymbol{{In}}\:\boldsymbol{{x}}\right)\mathrm{2}}{\boldsymbol{{x}}}\:\boldsymbol{{dx}}\:=\:\frac{\mathrm{1}}{\mathrm{3}}\:\left(\boldsymbol{{In}}\:\boldsymbol{{x}}\right)^{\mathrm{3}} \\ $$ Commented by som(math1967) last updated on 14/Jan/22 $$\int\left({lnx}\right)^{\mathrm{2}} {d}\left({lnx}\right)=\frac{\mathrm{1}}{\mathrm{3}}\left({lnx}\right)^{\mathrm{3}}…

very-nice-to-problem-find-in-closed-form-0-1-log-1-x-2-log-n-1-x-dx-n-N-z-

Question Number 164123 by Zaynal last updated on 14/Jan/22 $$\mathrm{very}\:\mathrm{nice}\:\mathrm{to}\:\mathrm{problem}: \\ $$$$\boldsymbol{{find}}\:\boldsymbol{{in}}\:\boldsymbol{{closed}}\:\boldsymbol{{form}}; \\ $$$$\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\boldsymbol{{log}}\:\left(\mathrm{1}−\boldsymbol{{x}}^{\mathrm{2}} \right)\:\boldsymbol{{log}}\:^{\boldsymbol{{n}}\:} \:\left(\mathrm{1}−\boldsymbol{{x}}\right)\:\boldsymbol{{dx}}; \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{n}}\:\in\:\:\mathbb{N}^{+} \\ $$$$\:^{\mathrm{z}.} \\ $$ Terms…

help-dx-csc-x-1-my-way-dx-1-sinx-1-sinx-1-sinx-dx-sinx-1-1-sinx-1-dx-1-1-sinx-1-dx-1dx-sinx-1-sinx-1-sinx-1-dx-x-C-si

Question Number 33009 by kyle_TW last updated on 09/Apr/18 $${help}\:!\:!\:! \\ $$$$\int\:\frac{{dx}}{{csc}\left({x}\right)−\mathrm{1}}\:=\:? \\ $$$$ \\ $$$$\left[\:{my}\:{way}\:\right] \\ $$$$\int\left(\:\frac{{dx}}{\frac{\mathrm{1}}{{sinx}}\:−\:\mathrm{1}}\:\right) \\ $$$$=\int\frac{{sinx}}{\mathrm{1}−{sinx}}\:{dx} \\ $$$$=−\int\:\frac{{sinx}−\mathrm{1}+\mathrm{1}}{{sinx}−\mathrm{1}}\:{dx} \\ $$$$=−\int\mathrm{1}+\frac{\mathrm{1}}{{sinx}−\mathrm{1}}\:{dx} \\…