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Category: Integration

nice-calculus-prove-that-1-0-1-li-2-1-x-2-pi-2-2-4-2-0-1-log-1-t-t-3-4-1-t-dt-pi-3-2-2-2-3-4-log-2-pi-2-hi

Question Number 131991 by mnjuly1970 last updated on 10/Feb/21 $$\:\:\:\:\:\:\:\:\:\:\:\:…{nice}\:\:\:\:\:\:\:\:\:{calculus}…\:\: \\ $$$$\:\:{prove}\:\:{that}\:: \\ $$$$\:\:\phi_{\mathrm{1}} =\int_{\mathrm{0}} ^{\:\mathrm{1}} {li}_{\mathrm{2}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)=\frac{\pi^{\mathrm{2}} }{\mathrm{2}}\:−\mathrm{4} \\ $$$$\:\:\:\:\phi_{\mathrm{2}} =\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{log}\left(\mathrm{1}−{t}\right)}{{t}^{\frac{\mathrm{3}}{\mathrm{4}}}…

Determine-the-results-of-the-following-I-1-a-e-x-a-e-x-dx-I-2-tanx-tanx-1-tan-2-x-dx-

Question Number 907 by Yugi last updated on 19/Apr/15 $${Determine}\:{the}\:{results}\:{of}\:{the}\:{following}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{I}_{\mathrm{1}} =\int\sqrt{\frac{{a}+{e}^{{x}} }{{a}−{e}^{{x}} }}{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{I}_{\mathrm{2}} =\int\frac{\left({tanx}\right)\mid{tanx}\mid}{\mathrm{1}−{tan}^{\mathrm{2}} {x}}{dx} \\ $$ Commented by prakash jain…

math-analysis-xsin-x-x-2-2x-2-dx-xsin-x-x-1-2-1-dx-x-1-t-t-1-sin-t-1-t-2-1-dt-

Question Number 131969 by mnjuly1970 last updated on 10/Feb/21 $$\:\:\:\:\:\:\:\:\:\:\:\:…\:{math}\:\:{analysis}… \\ $$$$\:\:\:\:\phi=\:\:\int_{−\infty} ^{\:+\infty} \frac{{xsin}\left({x}\right)}{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{2}}{dx}=? \\ $$$$\:\:\:\:\:\phi=\int_{−\infty} ^{\:+\infty} \frac{{xsin}\left({x}\right)}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} +\mathrm{1}}{dx} \\ $$$$\:\:\:\:\:\:\:\overset{{x}+\mathrm{1}={t}} {=}\int_{−\infty} ^{\:+\infty} \frac{\left({t}−\mathrm{1}\right){sin}\left({t}−\mathrm{1}\right)}{{t}^{\mathrm{2}}…

sin-4-x-cos-4-x-dx-

Question Number 131961 by Zack_ last updated on 10/Feb/21 $$\int\left({sin}^{\mathrm{4}} {x}.{cos}^{\mathrm{4}} {x}\right){dx} \\ $$ Answered by liberty last updated on 10/Feb/21 $$\mathrm{I}=\int\:\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:\mathrm{2x}\right)^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:\mathrm{2x}\right)^{\mathrm{2}} \mathrm{dx} \\…