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Category: Integration

answer-to-26024-let-put-c-0-cos-ax-2-dx-and-c-0-sin-ax-2-dx-ew-have-c-is-0-e-iax-2-dx-2-1-R-e-iax-2-dx-and-i-put-x-1-2-r-x-notation-

Question Number 26107 by abdo imad last updated on 19/Dec/17 $${answer}\:{to}\:\mathrm{26024}\:\:\:\:{let}\:\:{put}\:{c}=\:\int_{\mathrm{0}} ^{\infty} \:{cos}\left({ax}^{\mathrm{2}} \right){dx}\:\:\:{and}\:\:\:{c}\:=\:\:\int_{\mathrm{0}} ^{\infty} \:{sin}\left({ax}^{\mathrm{2}} \right){dx} \\ $$$${ew}\:{have}\:\:{c}−{is}\:\:=\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{iax}^{\mathrm{2}} } {dx}\:\:\:=\mathrm{2}^{−\mathrm{1}} \:\:\int_{{R}} \:{e}^{−{iax}^{\mathrm{2}}…

show-that-pi-4-pi-4-sin2x-2-cos2x-2-ln-1-e-x-dx-pi-16-pi-3-36-

Question Number 91630 by Ar Brandon last updated on 02/May/20 $$\mathrm{show}\:\mathrm{that}\: \\ $$$$\int_{−\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{4}}} \frac{\mathrm{sin2x}}{\left(\mathrm{2}+\mathrm{cos2x}\right)^{\mathrm{2}} }\mathrm{ln}\left(\mathrm{1}+\mathrm{e}^{\mathrm{x}} \right)\mathrm{dx} \\ $$$$=\frac{\pi}{\mathrm{16}}−\frac{\pi\sqrt{\mathrm{3}}}{\mathrm{36}} \\ $$ Commented by Ar Brandon…

asin-3-bcos-3-sin-2-cos-2-d-

Question Number 26090 by fahim last updated on 19/Dec/17 $$\int\frac{{a}\mathrm{sin}\:^{\mathrm{3}} \theta+{b}\mathrm{cos}\:^{\mathrm{3}} \theta}{\mathrm{sin}^{\mathrm{2}} \:\theta.\mathrm{cos}^{\mathrm{2}} \:\theta}{d}\theta \\ $$ Answered by Joel578 last updated on 19/Dec/17 $${I}\:=\:\int\:\frac{{a}\mathrm{sin}^{\mathrm{3}} \:\theta}{\mathrm{sin}^{\mathrm{2}}…

let-f-x-2-x-x-1-find-f-x-f-1-x-dx-and-ln-f-x-f-1-x-dx-

Question Number 91620 by abdomathmax last updated on 02/May/20 $${let}\:{f}\left({x}\right)\:=\mathrm{2}\:{x}−\sqrt{{x}−\mathrm{1}} \\ $$$${find}\:\int\:\:\frac{{f}\left({x}\right)}{{f}^{−\mathrm{1}} \left({x}\right)}{dx}\:\:{and}\:\:\int\:{ln}\left(\frac{{f}\left({x}\right)}{{f}^{−\mathrm{1}} \left({x}\right)}\right){dx} \\ $$ Commented by mathmax by abdo last updated on 02/May/20…