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Category: Integration

siny-y-dy-

Question Number 67359 by mhmd last updated on 26/Aug/19 $$\int{siny}/{y}\:\:{dy} \\ $$ Commented by mathmax by abdo last updated on 26/Aug/19 $${at}\:{form}\:{of}\:{serie}\:\:{we}\:{have}\:{siny}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} {y}^{\mathrm{2}{n}+\mathrm{1}}…

nice-calculus-prove-that-0-sin-x-log-2-x-x-pi-24-12-2-pi-2-

Question Number 132877 by mnjuly1970 last updated on 17/Feb/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:….\:{nice}\:\:{calculus}… \\ $$$$\:\:\:{prove}\:{that}\:: \\ $$$$\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left({x}\right).{log}^{\mathrm{2}} \left({x}\right)}{{x}} \\ $$$$\:\:\:\:\:\:\:=\frac{\pi}{\mathrm{24}}\left(\mathrm{12}\gamma^{\:\mathrm{2}} +\pi^{\mathrm{2}} \right)\:….. \\ $$ Answered by…

Let-P-t-denote-a-given-cubic-polynomial-Find-the-constants-a-1-u-1-a-2-and-u-2-such-that-1-1-P-t-dt-a-1-P-u-1-a-2-P-u-2-

Question Number 1778 by 112358 last updated on 23/Sep/15 $${Let}\:{P}\left({t}\right)\:{denote}\:{a}\:{given}\:{cubic} \\ $$$${polynomial}.\:{Find}\:{the}\:{constants} \\ $$$${a}_{\mathrm{1}} ,{u}_{\mathrm{1}} ,{a}_{\mathrm{2}} \:{and}\:{u}_{\mathrm{2}} \:{such}\:{that} \\ $$$$\int_{−\mathrm{1}} ^{\:\mathrm{1}} {P}\left({t}\right){dt}={a}_{\mathrm{1}} {P}\left({u}_{\mathrm{1}} \right)+{a}_{\mathrm{2}} {P}\left({u}_{\mathrm{2}}…