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Category: Integration

Prove-that-0-pi-2-sin-2-x-sin-x-cos-x-dx-1-2-log-2-1-

Question Number 177298 by peter frank last updated on 03/Oct/22 $$\mathrm{Prove}\:\mathrm{that} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}}{\left(\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}\right)}\mathrm{dx}=\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\mathrm{log}\:\left(\sqrt{\mathrm{2}}+\mathrm{1}\right) \\ $$ Answered by Ar Brandon last updated on…

Question-46225

Question Number 46225 by rahul 19 last updated on 22/Oct/18 Answered by MrW3 last updated on 23/Oct/18 $${let}\:{f}\left({x}\right)={a}\left({x}−{b}\right)^{\mathrm{2}} +{c} \\ $$$$\left({b}\right)\Rightarrow{b}=\mathrm{1},\:{c}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left({a}\right)\Rightarrow{a}\left(\mathrm{0}−\mathrm{1}\right)^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}=\mathrm{0}\Rightarrow{a}=−\frac{\mathrm{1}}{\mathrm{2}} \\…

Question-46221

Question Number 46221 by Meritguide1234 last updated on 22/Oct/18 Answered by MJS last updated on 22/Oct/18 $$\mathrm{I}\:\mathrm{solved}\:\mathrm{it}\:\mathrm{but}\:\mathrm{I}'\mathrm{m}\:\mathrm{too}\:\mathrm{tired}\:\mathrm{to}\:\mathrm{type}\:\mathrm{it} \\ $$ Commented by Meritguide1234 last updated on…

Using-dimensional-analysis-find-out-value-of-n-in-given-expression-dx-2ax-x-2-a-n-sin-1-x-a-1-

Question Number 46188 by rahul 19 last updated on 22/Oct/18 $${Using}\:{dimensional}\:{analysis}\:, \\ $$$${find}\:{out}\:{value}\:{of}\:{n}\:{in}\:{given}\:{expression}: \\ $$$$\:\:\int\frac{{dx}}{\:\sqrt{\mathrm{2}{ax}−{x}^{\mathrm{2}} }}\:=\:{a}^{{n}} \mathrm{sin}^{−\mathrm{1}} \left(\frac{{x}}{{a}}\:−\mathrm{1}\right). \\ $$ Commented by rahul 19 last…

Question-46182

Question Number 46182 by Meritguide1234 last updated on 22/Oct/18 Commented by maxmathsup by imad last updated on 22/Oct/18 $$\:{we}\:{have}\:\mathrm{1}−{x}^{\mathrm{2}} \:+{x}^{\mathrm{4}} −….=\sum_{{n}=\mathrm{0}} ^{\infty} \left(−{x}^{\mathrm{2}} \right)^{{n}} \:=\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}}…

advanced-mathematics-please-demonstrate-that-0-1-xlog-1-x-log-1-x-1-4-log-2-m-n-july-1970-

Question Number 111719 by mnjuly1970 last updated on 04/Sep/20 $$\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:….{advanced}\:\:{mathematics}….\: \\ $$$$ \\ $$$${please}\:\:{demonstrate}\:{that}:: \\ $$$$\: \\ $$$$\Phi\:=\int_{\mathrm{0}} ^{\:\mathrm{1}} {xlog}\left(\mathrm{1}−{x}\right).{log}\left(\mathrm{1}+{x}\right)=\:\frac{\mathrm{1}}{\mathrm{4}}\:−\:{log}\left(\mathrm{2}\right)\:\:… \\ $$$$ \\…